Step-by-Step Guidance for Introductory Statistics Worksheet

{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. What is the probability that a randomly selected card from a standard deck of 52 cards is Red or Diamond?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Basic Probability, Counting Rules"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of probability with mutually inclusive events and how to count outcomes in a standard deck of cards."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Red cards: Hearts and Diamonds (total of 26 cards)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Diamond cards: 13 cards"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability formula: "},{"type":"inlineMath","attrs":{"latex":"P(A \\text{ or } B) = P(A) + P(B) - P(A \\text{ and } B)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total cards: 52"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the number of red cards in the deck (Hearts and Diamonds)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the number of Diamond cards in the deck."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Determine the overlap: Diamonds are red, so all Diamond cards are already counted as red."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the formula for probability of \"Red or Diamond\" using "},{"type":"inlineMath","attrs":{"latex":"P(A \\text{ or } B) = P(A) + P(B) - P(A \\text{ and } B)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation for the probability, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{26}{52} = 0.5"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"All Diamonds are red, so the probability is simply the probability of drawing a red card: "},{"type":"inlineMath","attrs":{"latex":"\\frac{26}{52} = 0.5"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2a. Complete the table for adult and children's books by fiction/nonfiction."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Contingency Tables, Counting"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to organize categorical data into a two-way table."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Adult books: 100 total, 70 nonfiction"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Children's books: 160 total, 100 fiction"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Table categories: Fiction, Nonfiction, Totals"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Fill in the known values: Adult nonfiction (70), Children's fiction (100)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate Adult fiction: "},{"type":"inlineMath","attrs":{"latex":"100 - 70 = 30"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate Children's nonfiction: "},{"type":"inlineMath","attrs":{"latex":"160 - 100 = 60"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sum each column and row to get totals."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the completed table, but do not reveal all totals yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null}}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Fiction"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Nonfiction"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Adult Books"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"30"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"70"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"100"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Children's Books"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"100"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"60"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"160"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"130"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"130"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"260"}]}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2b. What is the probability that a randomly selected book is a fiction book?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Probability from Contingency Tables"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find probabilities from categorical data."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total fiction books: sum from table"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total books: sum from table"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability formula: "},{"type":"inlineMath","attrs":{"latex":"P = \\frac{\\text{Number of fiction books}}{\\text{Total number of books}}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the total number of fiction books from the completed table."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the total number of books."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the probability calculation using the formula above."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{130}{260} = 0.5"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are 130 fiction books out of 260 total books, so the probability is "},{"type":"inlineMath","attrs":{"latex":"0.5"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2c. What is the probability that a randomly selected book is a nonfiction book and a children's book?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Joint Probability"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the probability of two simultaneous events using a contingency table."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Children's nonfiction books: from table"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total books: from table"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability formula: "},{"type":"inlineMath","attrs":{"latex":"P = \\frac{\\text{Number of children's nonfiction books}}{\\text{Total number of books}}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the number of children's nonfiction books from the table."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the total number of books."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the probability calculation using the formula above."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{60}{260} \\approx 0.23"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are 60 children's nonfiction books out of 260 total books, so the probability is approximately "},{"type":"inlineMath","attrs":{"latex":"0.23"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2d. What is the probability that a randomly selected book is a fiction book or an adult book?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Probability of Union of Two Events"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the formula for the probability of \"A or B\" with overlapping categories."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Adult books: 100"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Fiction books: 130"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Adult fiction books: 30"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability formula: "},{"type":"inlineMath","attrs":{"latex":"P(A \\text{ or } B) = P(A) + P(B) - P(A \\text{ and } B)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the number of adult books, fiction books, and adult fiction books from the table."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the formula for probability of \"fiction or adult\" using "},{"type":"inlineMath","attrs":{"latex":"P(A \\text{ or } B) = P(A) + P(B) - P(A \\text{ and } B)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation for the probability, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{100 + 130 - 30}{260} = \\frac{200}{260} \\approx 0.77"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Using the formula, the probability is approximately "},{"type":"inlineMath","attrs":{"latex":"0.77"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3a. Construct the probability distribution for the number of TVs in a household."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Probability Distributions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to create a probability distribution from frequency data."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability distribution: List each value and its probability"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability formula: "},{"type":"inlineMath","attrs":{"latex":"P(x) = \\frac{\\text{Frequency of } x}{\\text{Total frequency}}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sum all frequencies to find the total number of people surveyed."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each number of TVs (0 to 7), calculate the probability by dividing its frequency by the total."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the probability distribution table, but do not fill in all probabilities yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Number of TVs"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"0"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{9}{115}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"1"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{25}{115}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"2"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{29}{115}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"3"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{19}{115}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"4"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{15}{115}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"5"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{8}{115}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"6"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{6}{115}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"7"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{4}{115}"}}]}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3b. Find and interpret the mean of the data (number of TVs)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Mean of a Probability Distribution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to calculate the expected value (mean) from a probability distribution."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Mean formula: "},{"type":"inlineMath","attrs":{"latex":"\\mu = \\sum x P(x)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply each number of TVs ("},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":") by its probability "},{"type":"inlineMath","attrs":{"latex":"P(x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sum all these products to get the mean."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\mu \\approx 2.49"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The mean number of TVs per household is about 2.49. This means that, on average, each household has about 2.5 TVs."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3c. Find and interpret the standard deviation of the data (number of TVs)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Standard Deviation of a Probability Distribution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to calculate the spread of values in a probability distribution."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Standard deviation formula: "},{"type":"inlineMath","attrs":{"latex":"\\sigma = \\sqrt{\\sum (x - \\mu)^2 P(x)}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract the mean from each value of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" and square the result."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply each squared difference by its probability "},{"type":"inlineMath","attrs":{"latex":"P(x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sum all these products and take the square root."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\sigma \\approx 1.49"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The standard deviation is about 1.49 TVs, indicating the typical variation from the mean number of TVs per household."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4a. What are the possible values of X, the number of students who graduated high school in a sample of 10?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Binomial Distribution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the possible outcomes in a binomial experiment."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Binomial random variable: "},{"type":"inlineMath","attrs":{"latex":"X"}},{"type":"text","text":" can take integer values from 0 to "},{"type":"inlineMath","attrs":{"latex":"n"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Here, "},{"type":"inlineMath","attrs":{"latex":"n = 10"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"List all possible values for "},{"type":"inlineMath","attrs":{"latex":"X"}},{"type":"text","text":" in a binomial experiment with "},{"type":"inlineMath","attrs":{"latex":"n = 10"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the range, but do not list all values yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"X = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"These are all possible numbers of students who could graduate in a sample of 10."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4b. What is the probability that exactly three students graduated high school?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Binomial Probability"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the binomial probability formula."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Binomial probability formula: "},{"type":"inlineMath","attrs":{"latex":"P(X = k) = \\binom{n}{k} p^k (1-p)^{n-k}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"n = 10"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"k = 3"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"p = 0.84"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"k"}},{"type":"text","text":", and "},{"type":"inlineMath","attrs":{"latex":"p"}},{"type":"text","text":" for the problem."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plug these values into the binomial formula."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"P(X = 3) \\approx 0.0009"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The probability that exactly three students graduated is about 0.0009."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4c. What is the probability that fewer than seven students graduated high school?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Cumulative Binomial Probability"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to sum binomial probabilities for "},{"type":"inlineMath","attrs":{"latex":"X < 7"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Cumulative probability: "},{"type":"inlineMath","attrs":{"latex":"P(X < 7) = \\sum_{k=0}^{6} P(X = k)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use binomial formula for each "},{"type":"inlineMath","attrs":{"latex":"k"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"k = 0"}},{"type":"text","text":" to $6"},{"type":"inlineMath","attrs":{"latex":", calculate "}},{"type":"text","text":"P(X = k)$ using the binomial formula."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sum all these probabilities to get "},{"type":"inlineMath","attrs":{"latex":"P(X < 7)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"P(X < 7) \\approx 0.003"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The probability that fewer than seven students graduated is about 0.003."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4d. What is the probability that at least four students graduated high school?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Cumulative Binomial Probability"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to sum binomial probabilities for "},{"type":"inlineMath","attrs":{"latex":"X \\geq 4"}},{"type":"text","text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"P(X \\geq 4) = \\sum_{k=4}^{10} P(X = k)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Alternatively, "},{"type":"inlineMath","attrs":{"latex":"P(X \\geq 4) = 1 - P(X < 4)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate "},{"type":"inlineMath","attrs":{"latex":"P(X < 4)"}},{"type":"text","text":" by summing "},{"type":"inlineMath","attrs":{"latex":"P(X = 0)"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"P(X = 1)"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"P(X = 2)"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"P(X = 3)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract this sum from 1 to get "},{"type":"inlineMath","attrs":{"latex":"P(X \\geq 4)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"P(X \\geq 4) \\approx 0.999"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The probability that at least four students graduated is about 0.999."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4e. Compute the mean and interpret for the binomial distribution."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Mean of Binomial Distribution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the expected value for a binomial random variable."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Mean formula: "},{"type":"inlineMath","attrs":{"latex":"\\mu = n p"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"n = 10"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"p = 0.84"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plug "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"p"}},{"type":"text","text":" into the mean formula."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\mu = 8.4"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"On average, 8.4 out of 10 students are expected to graduate high school."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4f. Compute the standard deviation and interpret for the binomial distribution."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Standard Deviation of Binomial Distribution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the spread for a binomial random variable."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Standard deviation formula: "},{"type":"inlineMath","attrs":{"latex":"\\sigma = \\sqrt{n p (1-p)}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"n = 10"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"p = 0.84"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plug "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"p"}},{"type":"text","text":" into the standard deviation formula."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\sigma \\approx 1.29"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The standard deviation is about 1.29, meaning the number of graduates typically varies by about 1.3 from the mean."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5a. Construct a probability distribution for the raffle (winning $500 or losing $1)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Probability Distribution for Discrete Outcomes"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to model a simple game of chance with two outcomes."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Winning: $500"},{"type":"inlineMath","attrs":{"latex":" prize, probability "}},{"type":"text","text":"\\frac{1}{1000}$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Losing: "},{"type":"inlineMath","attrs":{"latex":"-1"}},{"type":"text","text":" (cost of ticket), probability "},{"type":"inlineMath","attrs":{"latex":"\\frac{999}{1000}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the two possible outcomes: win $500"},{"type":"inlineMath","attrs":{"latex":" or lose $1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the probability for each outcome."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the probability distribution table, but do not fill in all probabilities yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Outcome"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"$500$"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{1}{1000}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"-1"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{999}{1000}"}}]}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5b. Determine the expected value for a person who buys one ticket."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Expected Value"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to calculate the expected value from a probability distribution."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expected value formula: "},{"type":"inlineMath","attrs":{"latex":"E(X) = \\sum x P(x)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply each outcome by its probability."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sum these products to get the expected value."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"E(X) = (500) \\frac{1}{1000} + (-1) \\frac{999}{1000} = -0.499"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The expected value is about "},{"type":"inlineMath","attrs":{"latex":"-0.50"}},{"type":"text","text":", meaning you lose about 50 cents per ticket on average."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6a. Construct a probability distribution for betting $1 on a number in roulette (win $35 or lose $1)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Probability Distribution for Gambling"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to model a game with two outcomes and payoff odds."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Win: $35"},{"type":"inlineMath","attrs":{"latex":" payoff, probability "}},{"type":"text","text":"\\frac{1}{38}$ (American roulette)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Lose: "},{"type":"inlineMath","attrs":{"latex":"-1"}},{"type":"text","text":", probability "},{"type":"inlineMath","attrs":{"latex":"\\frac{37}{38}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the two possible outcomes: win $35"},{"type":"inlineMath","attrs":{"latex":" or lose $1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the probability for each outcome."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the probability distribution table, but do not fill in all probabilities yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Outcome"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"$35$"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{1}{38}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"-1"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null,"align":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{37}{38}"}}]}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6b. What is the expected payoff in roulette if you bet $1 on any number?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Expected Value in Gambling"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to calculate expected value from a probability distribution."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expected value formula: "},{"type":"inlineMath","attrs":{"latex":"E(X) = \\sum x P(x)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply each outcome by its probability."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sum these products to get the expected value."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"E(X) = (35) \\frac{1}{38} + (-1) \\frac{37}{38} \\approx -0.05"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The expected payoff is about "},{"type":"inlineMath","attrs":{"latex":"-0.05"}},{"type":"text","text":", meaning you lose about 5 cents per $1 bet on average."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. What is the probability that a randomly selected TV will be replaced within the first 5 years?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Normal Distribution, Probability"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the normal distribution to find probabilities for values below a given threshold."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Mean ("},{"type":"inlineMath","attrs":{"latex":"\\mu"}},{"type":"text","text":"): 8.2 years"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Standard deviation ("},{"type":"inlineMath","attrs":{"latex":"\\sigma"}},{"type":"text","text":"): 1.1 years"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Z-score formula: "},{"type":"inlineMath","attrs":{"latex":"z = \\frac{x - \\mu}{\\sigma}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability: "},{"type":"inlineMath","attrs":{"latex":"P(X < 5)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the z-score for "},{"type":"inlineMath","attrs":{"latex":"x = 5"}},{"type":"text","text":" using the formula."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the z-score to find the probability from the standard normal table."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"P(X < 5) \\approx 0.006"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The probability is about 0.6% that a TV will be replaced within the first 5 years."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8. Find the lowest possible score to qualify for Calculus (top 15%) on a placement test with mean 50 and SD 4."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Normal Distribution, Percentiles"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find a cutoff value corresponding to a percentile in a normal distribution."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Mean ("},{"type":"inlineMath","attrs":{"latex":"\\mu"}},{"type":"text","text":"): 50"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Standard deviation ("},{"type":"inlineMath","attrs":{"latex":"\\sigma"}},{"type":"text","text":"): 4"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Percentile: Top 15% ("},{"type":"inlineMath","attrs":{"latex":"P(X > x)"}},{"type":"text","text":")"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Z-score formula: "},{"type":"inlineMath","attrs":{"latex":"z = \\frac{x - \\mu}{\\sigma}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the z-score that corresponds to the top 15% (use normal table or calculator)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the equation "},{"type":"inlineMath","attrs":{"latex":"z = \\frac{x - 50}{4}"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final value yet."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"x \\approx 54"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The lowest possible score to qualify is about 54."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9a. Describe the sampling distribution of the sample mean for weights of 1-year-old boys (sample size 10)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Sampling Distribution, Central Limit Theorem"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the properties of the sampling distribution of the mean."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Population mean ("},{"type":"inlineMath","attrs":{"latex":"\\mu"}},{"type":"text","text":"): 22.8 lbs"}]}]},{"type":"listItem",