Step-by-Step Guidance for Math 111 Exam 2 Practice (Statistics)

Q1. What is the mean and standard deviation of the Standard Normal random variable Z?

Background

Topic: Standard Normal Distribution

This question tests your knowledge of the properties of the standard normal distribution, which is a fundamental concept in statistics.

Key Terms and Formulas:

• Standard Normal Distribution (Z): A normal distribution with mean $\mu = 0$ and standard deviation $\sigma = 1$.

• Mean ($\mu$): The center or average of the distribution.

• Standard Deviation ($\sigma$): A measure of the spread or variability of the distribution.

Step-by-Step Guidance

1. Recall the definition of the standard normal distribution. By definition, it is a normal distribution with specific parameters.

2. Identify the mean ($\mu$) of the standard normal distribution.

3. Identify the standard deviation ($\sigma$) of the standard normal distribution.

Try solving on your own before revealing the answer!

Q2. Identify which of the following statistics statements are incorrect or use notation erroneously.

Background

Topic: Statistical Notation and Interpretation

This question tests your ability to recognize correct and incorrect uses of statistical notation and concepts.

Key Terms and Formulas:

• Sample Mean ($\bar{x}$): The average of a sample.

• Population Mean ($\mu$): The average of a population.

• z-score: The number of standard deviations a value is from the mean.

• Probability Notation ($Pr$): $Pr(X > a)$ means the probability that $X$ is greater than $a$.

Step-by-Step Guidance

1. Read each statement carefully and recall the correct statistical notation and meaning.

2. For each statement, check if the notation matches standard usage (e.g., does $\bar{x}$ always equal $\mu$?).

3. For probability statements, ensure the right side of the equation makes sense (e.g., is $Pr(X > a) = 1 - a$ correct?).

4. For z-score and probability values, check if the notation and values are used appropriately.

Try solving on your own before revealing the answer!

Q3. For each pair of events (A, B), determine if they are mutually exclusive.

Background

Topic: Probability - Mutually Exclusive Events

This question tests your understanding of mutually exclusive (disjoint) events, which cannot occur at the same time.

Key Terms and Formulas:

• Mutually Exclusive Events: Two events are mutually exclusive if $P(A \cap B) = 0$ (they cannot both happen at once).

Step-by-Step Guidance

1. For each scenario, define what events A and B represent.

2. Consider if it is possible for both A and B to occur simultaneously.

3. If there is any overlap (i.e., a single outcome that satisfies both A and B), then they are not mutually exclusive.

4. If there is no overlap, then they are mutually exclusive.

Try solving on your own before revealing the answer!

Q4. How many different ways can you select one manufacturer, one car size, and one color from the given table?

Background

Topic: Counting Principle (Multiplication Rule)

This question tests your ability to use the multiplication rule to count the number of possible combinations.

Key Terms and Formulas:

• Multiplication Rule: If there are $m$ ways to do one thing, $n$ ways to do another, and $p$ ways to do a third, then there are $m \times n \times p$ ways to do all three.

Step-by-Step Guidance

1. Count the number of options for each category: manufacturer, car size, and color.

2. Multiply the number of choices for each category together to get the total number of combinations.

Try solving on your own before revealing the answer!

Q5. What is the probability of drawing a 5-card poker hand that contains AKQJ10 all of the same suit?

Background

Topic: Probability - Poker Hands, Combinatorics

This question tests your ability to calculate probabilities using combinations, specifically for special poker hands (a straight flush).

Key Terms and Formulas:

• Combination Formula: $C(n, k) = \frac{n!}{k!(n-k)!}$

• Probability: $P = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Step-by-Step Guidance

1. Determine how many ways you can get the hand AKQJ10 all in the same suit (i.e., how many suits are there?).

2. Recall the total number of possible 5-card hands from a 52-card deck (given in the problem).

3. Set up the probability as the ratio of the number of favorable hands to the total number of hands.

Try solving on your own before revealing the answer!

Q6. Given company employee data, compute the probabilities for (a) the employee is either male or left handed (or both), and (b) the employee is either a right handed male or a left handed female.

Background

Topic: Probability - Addition Rule, Venn Diagrams

This question tests your ability to use the addition rule for probabilities, including the concept of overlapping events.

Key Terms and Formulas:

• Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• Complement Rule: $P(A^c) = 1 - P(A)$

Step-by-Step Guidance

1. Identify the probabilities given: $P(\text{left handed}) = 0.15$, $P(\text{left handed and male}) = 0.06$, and the company has equal numbers of males and females.

2. For part (a), use the addition rule to find $P(\text{male} \cup \text{left handed})$.

3. For part (b), recognize that "right handed male" and "left handed female" are mutually exclusive, so add their probabilities.

4. Calculate the probability of being a right handed male and a left handed female using the given data and the complement rule.

Try solving on your own before revealing the answer!

Q7. How many different Virginia license plates are possible with four letters followed by three digits?

Background

Topic: Counting Principle (Multiplication Rule)

This question tests your ability to count the number of possible outcomes using the multiplication rule.

Key Terms and Formulas:

• Multiplication Rule: If there are $n$ ways to do one thing and $m$ ways to do another, there are $n \times m$ ways to do both.

• There are 26 letters in the English alphabet and 10 digits (0-9).

Step-by-Step Guidance

1. Determine the number of choices for each letter and each digit position.

2. Multiply the number of choices for all positions together to get the total number of license plates.

Try solving on your own before revealing the answer!

Q8. Is the given table a legitimate probability distribution for the random variable X (number of classes)?

Background

Topic: Probability Distributions

This question tests your ability to recognize the properties of a valid probability distribution.

Key Terms and Formulas:

• Probability Distribution: A table or function that gives the probability of each possible value of a random variable.

• Properties: All probabilities must be between 0 and 1, and the sum of all probabilities must equal 1.

Step-by-Step Guidance

1. Check that each probability is between 0 and 1.

2. Add up all the probabilities to see if they sum to 1.

3. If any probability is negative, greater than 1, or the sum is not 1, explain why the distribution is not legitimate.

Try solving on your own before revealing the answer!

Q9. Using the blood donor table, find the probability for (a) type O and Rh-negative, (b) type O or type A, (c) type B or Rh-negative.

Background

Topic: Probability - Contingency Tables

This question tests your ability to calculate probabilities from a two-way table.

Key Terms and Formulas:

• Joint Probability: $P(A \cap B) = \frac{\text{Number with both A and B}}{\text{Total}}$

• Union Probability: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Step-by-Step Guidance

1. For each part, identify the relevant counts from the table.

2. For joint probabilities, use the count for both characteristics divided by the total.

3. For union probabilities, add the relevant counts and subtract any overlap, then divide by the total.

Try solving on your own before revealing the answer!

Q10. What is the probability that a randomly selected card from a standard deck is between 4 and 6 (inclusive), or is a club?

Background

Topic: Probability - Inclusion-Exclusion Principle

This question tests your ability to use the inclusion-exclusion principle to find the probability of the union of two events.

Key Terms and Formulas:

• Inclusion-Exclusion Principle: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• There are 52 cards in a standard deck, 4 suits, and 13 denominations per suit.

Step-by-Step Guidance

1. Count the number of cards between 4 and 6 (inclusive) in the deck.

2. Count the number of clubs in the deck.

3. Count the number of cards that are both between 4 and 6 and are clubs.

4. Apply the inclusion-exclusion formula to find the probability.

Try solving on your own before revealing the answer!

Q11. Rolling two fair 8-sided dice: (a) How many elements in the sample space? (b) Probability that the sum is 16? (c) Probability that the sum is strictly greater than 14?

Background

Topic: Probability - Sample Spaces and Event Probabilities

This question tests your ability to enumerate sample spaces and calculate probabilities for specific outcomes.

Key Terms and Formulas:

• Sample Space Size: For two dice, $n^2$ where $n$ is the number of sides.

• Probability: $P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$

Step-by-Step Guidance

1. For (a), calculate the total number of possible outcomes when rolling two 8-sided dice.

2. For (b), determine all pairs of dice rolls that sum to 16.

3. For (c), determine all pairs of dice rolls that sum to more than 14.

4. For each probability, set up the ratio of favorable outcomes to total outcomes.

Try solving on your own before revealing the answer!

Q12. What is the probability that the mean sales price of 12 randomly selected homes is more than $275,000, given a normal distribution with mean $296,700 and standard deviation $50,000?

Background

Topic: Sampling Distributions, Central Limit Theorem

This question tests your ability to use the sampling distribution of the sample mean to calculate probabilities.

Key Terms and Formulas:

• Sampling Distribution of the Mean: $\mu_{\bar{x}} = \mu$, $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• z-score for sample mean: $z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}}$

Step-by-Step Guidance

1. Identify the population mean ($\mu$), population standard deviation ($\sigma$), and sample size ($n$).

2. Calculate the standard error of the mean: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$.

3. Compute the z-score for $\bar{x} = 275,000$ using the formula above.

4. Use the standard normal table to find the probability that $\bar{x} > 275,000$.

Try solving on your own before revealing the answer!

Q13. Write out the entire sample space for flipping a coin and rolling a 4-sided die.

Background

Topic: Sample Spaces, Basic Probability

This question tests your ability to enumerate all possible outcomes of a compound experiment.

Key Terms and Formulas:

• Sample Space: The set of all possible outcomes.

Step-by-Step Guidance

1. List the possible outcomes for the coin flip (e.g., Heads, Tails).

2. List the possible outcomes for the 4-sided die (numbers 1 through 4).

3. Combine each coin outcome with each die outcome to enumerate the full sample space.

Try solving on your own before revealing the answer!

Q14. Using the Standard Normal Table, find $Pr(Z < 0 \text{ or } Z > 1.68)$. Accompany your calculations with an appropriate picture.

Background

Topic: Standard Normal Probabilities, Area Under the Curve

This question tests your ability to use the standard normal table to find probabilities and interpret the results graphically.

Key Terms and Formulas:

• Standard Normal Table: Gives $P(Z < z)$ for various $z$ values.

• Union of Events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Step-by-Step Guidance

1. Find $P(Z < 0)$ using the standard normal table.

2. Find $P(Z > 1.68)$ by calculating $1 - P(Z < 1.68)$.

3. Add the two probabilities together (since the events are disjoint).

4. Draw a sketch of the standard normal curve, shading the appropriate regions.

Try solving on your own before revealing the answer!

Q15. For pregnancy length (mean 267 days, SD 10 days), find the probability that a randomly selected new mother has a pregnancy length between 260 and 300 days.

Background

Topic: Normal Distribution Probabilities

This question tests your ability to calculate probabilities for intervals under the normal distribution.

Key Terms and Formulas:

• z-score: $z = \frac{x - \mu}{\sigma}$

• Probability: $P(a < X < b) = P(Z < z_b) - P(Z < z_a)$

Step-by-Step Guidance

1. Calculate the z-scores for 260 and 300 days.

2. Use the standard normal table to find the probabilities corresponding to each z-score.

3. Subtract the lower probability from the higher to get the probability for the interval.

Try solving on your own before revealing the answer!

Q16. For cholesterol levels (mean 190, SD 40.9), find the highest value that is still in the bottom 1%.

Background

Topic: Normal Distribution Percentiles

This question tests your ability to find percentiles (cutoff values) for a normal distribution.

Key Terms and Formulas:

• Percentile: The value below which a given percentage of observations fall.

• z-score for percentile: Use the standard normal table to find the z-score corresponding to the 1st percentile.

• Value: $x = \mu + z \sigma$

Step-by-Step Guidance

1. Use the standard normal table to find the z-score corresponding to the bottom 1% (i.e., $P(Z < z) = 0.01$).

2. Plug the z-score, mean, and standard deviation into the formula $x = \mu + z \sigma$ to find the cutoff value.

Try solving on your own before revealing the answer!

Q17. Roulette wheel: (a) Probability that none of 3 spins are red? (b) Probability that at least 1 spin is red?

Background

Topic: Probability - Independent Events, Complements

This question tests your ability to calculate probabilities for independent events and use the complement rule.

Key Terms and Formulas:

• Probability of Red: $P(\text{Red}) = \frac{18}{38}$

• Probability of Not Red: $P(\text{Not Red}) = 1 - P(\text{Red})$

• Independent Events: $P(A \text{ and } B) = P(A) \times P(B)$

• Complement Rule: $P(\text{At least one Red}) = 1 - P(\text{No Red})$

Step-by-Step Guidance

1. Calculate the probability that a single spin is not red.

2. Raise this probability to the third power to find the probability that none of the three spins are red.

3. Use the complement rule to find the probability that at least one spin is red.

Try solving on your own before revealing the answer!

Q18. Let U be a uniform random variable on (1, 5): (a) Sketch the pdf, (b) Compute $P(1.25 < U < 3.5)$, (c) Compute $P(U = 2)$.

Background

Topic: Uniform Distribution (Continuous)

This question tests your understanding of the uniform distribution and how to compute probabilities for continuous random variables.

Key Terms and Formulas:

• Uniform pdf: $f(x) = \frac{1}{b - a}$ for $a < x < b$

• Probability for interval: $P(c < U < d) = \frac{d - c}{b - a}$

• Probability at a point (continuous): $P(U = k) = 0$

Step-by-Step Guidance

1. For (a), draw a rectangle from $x = 1$ to $x = 5$ with height $\frac{1}{4}$.

2. For (b), calculate the length of the interval $(1.25, 3.5)$ and divide by the total length $(5 - 1)$.

3. For (c), recall that the probability at a single point for a continuous random variable is zero.

Try solving on your own before revealing the answer!

Q19. How many different meals are available when you select an appetizer, an entree, and a dessert from 6, 12, and 8 options, respectively?

Background

Topic: Counting Principle (Multiplication Rule)

This question tests your ability to count the number of possible combinations using the multiplication rule.

Key Terms and Formulas:

• Multiplication Rule: $\text{Total} = 6 \times 12 \times 8$

Step-by-Step Guidance

1. Multiply the number of choices for appetizers, entrees, and desserts together.

Try solving on your own before revealing the answer!

Q20. Construct a probability distribution for Y = -1, 1, 2 and compute its expected value $E(Y)$.

Background

Topic: Discrete Probability Distributions, Expected Value

This question tests your ability to construct a probability distribution and calculate the expected value.

Key Terms and Formulas:

• Probability Distribution: Assign probabilities to each value so they sum to 1.

• Expected Value: $E(Y) = \sum y_i P(Y = y_i)$

Step-by-Step Guidance

1. Assign probabilities to each value of Y so that they sum to 1.

2. Multiply each value by its probability and sum the results to find the expected value.

Try solving on your own before revealing the answer!

Q21. For a sample of 45 chauffeurs (mean salary $30,800, SD $5,600), what is the probability that the sample mean is more than $30,000?

Background

Topic: Sampling Distributions, Central Limit Theorem

This question tests your ability to use the sampling distribution of the sample mean to calculate probabilities.

Key Terms and Formulas:

• Sampling Distribution of the Mean: $\mu_{\bar{x}} = \mu$, $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• z-score for sample mean: $z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}}$

Step-by-Step Guidance

1. Identify the population mean ($\mu$), population standard deviation ($\sigma$), and sample size ($n$).

2. Calculate the standard error of the mean: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$.

3. Compute the z-score for $\bar{x} = 30,000$ using the formula above.

4. Use the standard normal table to find the probability that $\bar{x} > 30,000$.

Try solving on your own before revealing the answer!

Q22. For $X \sim N(41, 12^2)$, (a) compute $P(20 < X < 50)$, (b) find the value marking the 9th percentile for X.

Background

Topic: Normal Distribution Probabilities and Percentiles

This question tests your ability to calculate probabilities and percentiles for a normal distribution.

Key Terms and Formulas:

• z-score: $z = \frac{x - \mu}{\sigma}$

• Probability: $P(a < X < b) = P(Z < z_b) - P(Z < z_a)$

• Percentile Value: $x = \mu + z \sigma$

Step-by-Step Guidance

1. For (a), calculate the z-scores for 20 and 50, then use the standard normal table to find the probabilities and subtract.

2. For (b), use the standard normal table to find the z-score for the 9th percentile, then solve for $x$ using $x = \mu + z \sigma$.

Try solving on your own before revealing the answer!