Statistics Hardest Questions: Advanced Placement Exam! Trivia Quiz

1. A university is proposing a new procedure for professors to gain tenure. To gauge sentiment about the proposal, the university intends to randomly sample five professors, five assistant professors, and five associate professors. This is an example of what type of sampling design?

Simple random sample

Stratified random sample

Systematic random sample

Cluster sample

Convenience sample

The university's proposal to randomly sample professors, assistant professors, and associate professors is an example of a stratified random sample. In this sampling design, the population is divided into distinct groups or strata (in this case, the different professor ranks), and a random sample is taken from each stratum. This ensures that each group is represented in the sample proportionally to its size in the population, allowing for more accurate conclusions about the sentiment towards the tenure proposal among different professor ranks.

Explanation

The university's proposal to randomly sample professors, assistant professors, and associate professors is an example of a stratified random sample. In this sampling design, the population is divided into distinct groups or strata (in this case, the different professor ranks), and a random sample is taken from each stratum. This ensures that each group is represented in the sample proportionally to its size in the population, allowing for more accurate conclusions about the sentiment towards the tenure proposal among different professor ranks.

2. A random sample of 25 birth-weights (in ounces) is taken, yielding the following summary statistics. What can be said about the number of outliers for this data set?

0

At least 1

No more than 1

At least 2

No more than 2

Based on the given information, it can be said that there is at least one outlier in the data set.

Explanation

Based on the given information, it can be said that there is at least one outlier in the data set.

3. A random variable has a standard deviation of 1.3. A new variable is created by transforming the values of the random variable using the following rule: Multiply each value by 2 and then add 5. What is the value of the standard deviation for this transformed variable?

1.3

2.6

6.3

7.6

8.5

When a random variable is transformed by multiplying each value by a constant and then adding another constant, the standard deviation of the transformed variable is equal to the product of the original standard deviation and the absolute value of the multiplying constant. In this case, the multiplying constant is 2, so the standard deviation of the transformed variable is 1.3 * 2 = 2.6.

Explanation

When a random variable is transformed by multiplying each value by a constant and then adding another constant, the standard deviation of the transformed variable is equal to the product of the original standard deviation and the absolute value of the multiplying constant. In this case, the multiplying constant is 2, so the standard deviation of the transformed variable is 1.3 * 2 = 2.6.

4. An airline has an on-time probability of 82.4%. What is the probability that, if you travel on this airline, no more than three of your next ten flights will not be on time? (Assume the flights are independent.) Select an answer.

A

B

C

D

E

The probability that no more than three out of ten flights will not be on time can be calculated using the binomial probability formula. The formula is P(X ≤ k) = Σ (n choose i) * p^i * (1-p)^(n-i), where n is the number of trials, i is the number of successes, p is the probability of success, and (n choose i) is the binomial coefficient. In this case, n = 10, i can be 0, 1, 2, or 3, p = 0.824, and (n choose i) can be calculated using the binomial coefficient formula. By plugging in these values and summing up the probabilities for i = 0, 1, 2, and 3, we can find the probability that no more than three flights will not be on time.

Explanation

The probability that no more than three out of ten flights will not be on time can be calculated using the binomial probability formula. The formula is P(X ≤ k) = Σ (n choose i) * p^i * (1-p)^(n-i), where n is the number of trials, i is the number of successes, p is the probability of success, and (n choose i) is the binomial coefficient. In this case, n = 10, i can be 0, 1, 2, or 3, p = 0.824, and (n choose i) can be calculated using the binomial coefficient formula. By plugging in these values and summing up the probabilities for i = 0, 1, 2, and 3, we can find the probability that no more than three flights will not be on time.

5. Twenty types of 12 oz. cans of soda were tested for calories and sugar (mg). The sodas averaged 156.85 calories with a standard deviation of 22.64, and the sugar level averaged 401.15 mg with a standard deviation of 102.43 mg. The correlation was given as r = 0.887. The equation of the LSRL predicting sugar level from calories is:

Predicted y = -228.30 + 4.01x

Predicted y = 22.64 - 156.85x

Predicted y = 102.43 - 401.15x

Predicted y = 4.01 - 228.33x

Predicted y = 78.19 + 0.196x

The equation of the least squares regression line (LSRL) is used to predict the value of the dependent variable (sugar level) based on the independent variable (calories). In this case, the LSRL equation is predicted y = -228.30 + 4.01x. This means that for every unit increase in calories, the predicted sugar level increases by 4.01 units. The intercept of -228.30 represents the predicted sugar level when the calories are zero.

Explanation

The equation of the least squares regression line (LSRL) is used to predict the value of the dependent variable (sugar level) based on the independent variable (calories). In this case, the LSRL equation is predicted y = -228.30 + 4.01x. This means that for every unit increase in calories, the predicted sugar level increases by 4.01 units. The intercept of -228.30 represents the predicted sugar level when the calories are zero.