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 the use of counting techniques with 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":"Diamonds: 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":"Count the number of red cards in the deck (Hearts and Diamonds)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of Diamond cards."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the overlap: Diamonds are red, so all Diamonds are already counted as red cards."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the formula for the 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 using the counts and the total number of cards, but do not compute the final probability 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 categorized by fiction and 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, Categorical Data"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to organize categorical data into a two-way table and understand marginal totals."}]},{"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":"Contingency table: A table showing the frequency distribution of variables."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Marginal totals: Totals for each row and column."}]}]}]},{"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 given values: 100 adult books, 160 children's books."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Of adult books, 70 are nonfiction. Use this to find the number of adult fiction books."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Of children's books, 100 are fiction. Use this to find the number of children's nonfiction books."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate row and column totals for fiction, nonfiction, adult, and children’s books."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the table with all categories, but leave the final numbers for you to fill in."}]}]}]},{"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":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Adult fiction: "},{"type":"inlineMath","attrs":{"latex":"100 - 70 = 30"}},{"type":"text","text":"; Children's nonfiction: "},{"type":"inlineMath","attrs":{"latex":"160 - 100 = 60"}},{"type":"text","text":"; Totals are calculated accordingly."}]}]},{"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 Frequency Tables"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to calculate probabilities from 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":"Probability: "},{"type":"inlineMath","attrs":{"latex":"P = \\frac{\\text{Number of favorable outcomes}}{\\text{Total outcomes}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Fiction books: Use the total from the table."}]}]}]},{"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 (adult + children)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the probability formula using these values."}]}]}]},{"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":"Joint probability: "},{"type":"inlineMath","attrs":{"latex":"P(A \\text{ and } B) = \\frac{\\text{Number in both categories}}{\\text{Total}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Children's nonfiction: Use the value from the table."}]}]}]},{"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 children’s nonfiction books from the 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 formula for this joint event."}]}]}]},{"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 Events"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the formula for the probability of the union of two events, accounting for overlap."}]},{"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 of union: "},{"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":"Adult books, fiction books, and adult fiction books: Use values from the table."}]}]}]},{"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 fiction books and adult books from the table."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the number of adult fiction books (overlap)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the formula for the probability of \"fiction or adult\" using the counts and total."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final probability 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{130 + 100 - 30}{260} = \\frac{200}{260} \\approx 0.77"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Fiction books: 130, adult books: 100, adult fiction: 30. Use the formula to get the probability."}]}]},{"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 based on the frequency data."}]},{"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, Empirical Probability"}]},{"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: A table showing the probability for each possible outcome."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Empirical probability: "},{"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), divide the frequency by the total to get the probability."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the probability 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Find and interpret the mean of the data (number of TVs per 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: Expected Value (Mean) of a Probability Distribution"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to calculate the mean of a discrete 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 (expected value): "},{"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":"Add up all these products to get the mean."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Interpret the mean in the context of the problem (average number of TVs per household)."}]}]}]},{"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 2.36"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The mean number of TVs per household is about 2.36, meaning that on average, each household has about 2.36 TVs."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3c. 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What are the possible values of "},{"type":"inlineMath","attrs":{"latex":"X"}},{"type":"text","marks":[{"type":"bold"}],"text":", 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":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Here, "},{"type":"inlineMath","attrs":{"latex":"n = 10"}},{"type":"text","text":"."}]}]}]},{"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":"Recall that in a binomial experiment with "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" trials, "},{"type":"inlineMath","attrs":{"latex":"X"}},{"type":"text","text":" can be any integer from 0 to $n$."}]}]},{"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":"."}]}]}]},{"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 = 0, 1, 2, ..., 10"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Possible values are all integers from 0 to 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: "},{"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 probability formula."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final probability 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) = \\binom{10}{3} (0.84)^3 (0.16)^7 \\approx 0.0006"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The probability is approximately 0.0006."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4c. 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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 calculate the standard deviation 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: "},{"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":"Calculate "},{"type":"inlineMath","attrs":{"latex":"n p (1-p)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Take the square root to find the standard deviation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Interpret the standard deviation in context (variation in number of graduates)."}]}]}]},{"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 = \\sqrt{10 \\times 0.84 \\times 0.16} \\approx 1.16"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The standard deviation is about 1.16, indicating the typical variation in the number of graduates in a sample of 10."}]}]},{"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 set up a probability distribution for a simple game of chance."}]},{"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 of winning: "},{"type":"inlineMath","attrs":{"latex":"P(\\text{win}) = \\frac{1}{1000}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability of losing: "},{"type":"inlineMath","attrs":{"latex":"P(\\text{lose}) = \\frac{999}{1000}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Outcomes: Win $500, Lose $1"}]}]}]},{"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 or lose."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Assign the monetary values to each outcome."}]}]},{"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."}]}]}]},{"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":"Win $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":"text","text":"Lose $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":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There is a 1 in 1000 chance to win $500, and a 999 in 1000 chance to lose $1."}]}]},{"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 expected value for a discrete 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: "},{"type":"inlineMath","attrs":{"latex":"E(X) = \\sum x P(x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the values and probabilities from the distribution."}]}]}]},{"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":"Add the products to get the expected value."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Interpret the expected value in context (average gain/loss per ticket)."}]}]}]},{"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 \\times \\frac{1}{1000}) + (-1 \\times \\frac{999}{1000}) = -0.499"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The expected value is about -$0.50, meaning on average, a person loses 50 cents per ticket."}]}]},{"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 Games of Chance"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to set up a probability distribution for a roulette bet."}]},{"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 of winning: "},{"type":"inlineMath","attrs":{"latex":"P(\\text{win}) = \\frac{1}{38}"}},{"type":"text","text":" (for American roulette)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability of losing: "},{"type":"inlineMath","attrs":{"latex":"P(\\text{lose}) = \\frac{37}{38}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Payoff: Win $35, Lose $1"}]}]}]},{"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 or lose."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Assign the monetary values to each outcome."}]}]},{"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."}]}]}]},{"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":"Win $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":"text","text":"Lose $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":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There is a 1 in 38 chance to win $35, and a 37 in 38 chance to lose $1."}]}]},{"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"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to calculate expected value for a game of chance."}]},{"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: "},{"type":"inlineMath","attrs":{"latex":"E(X) = \\sum x P(x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the values and probabilities from the distribution."}]}]}]},{"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":"Add the products to get the expected value."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Interpret the expected value in context (average gain/loss per bet)."}]}]}]},{"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 \\times \\frac{1}{38}) + (-1 \\times \\frac{37}{38}) = -0.0526"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The expected value is about -$0.05, meaning on average, you lose about 5 cents per $1 bet."}]}]},{"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? (Normal distribution: mean 8.2, SD 1.1)"}]},{"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: Normal Distribution, Probability Calculation"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the normal distribution to find probabilities for a given value."}]},{"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":"Normal distribution: "},{"type":"inlineMath","attrs":{"latex":"\\mu = 8.2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\sigma = 1.1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Z-score: "},{"type":"inlineMath","attrs":{"latex":"z = \\frac{x - \\mu}{\\sigma}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability: Use standard normal table for "},{"type":"inlineMath","attrs":{"latex":"z"}},{"type":"text","text":"."}]}]}]},{"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 "},{"type":"inlineMath","attrs":{"latex":"z"}},{"type":"text","text":"-score for "},{"type":"inlineMath","attrs":{"latex":"x = 5"}},{"type":"text","text":" using "},{"type":"inlineMath","attrs":{"latex":"z = \\frac{5 - 8.2}{1.1}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the standard normal table to find the probability corresponding to this "},{"type":"inlineMath","attrs":{"latex":"z"}},{"type":"text","text":"-score."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation, but do not compute the final probability 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 0.012"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The probability is about 0.012, meaning about 1.2% of TVs are replaced within 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 placement test, mean 50, SD 4, normal 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: Normal Distribution, Percentiles"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find a cutoff score 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":"Percentile: Top 15% means 85th percentile."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Z-score for 85th percentile: Use standard normal table."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Score: "},{"type":"inlineMath","attrs":{"latex":"x = \\mu + z \\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 "},{"type":"inlineMath","attrs":{"latex":"z"}},{"type":"text","text":"-score corresponding to the 85th percentile."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the formula "},{"type":"inlineMath","attrs":{"latex":"x = \\mu + z \\sigma"}},{"type":"text","text":" to find the cutoff score."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation,