AP Stats Chapter 12 Practice Test

Recent revenue shortfalls in a midwestern state led to a reduction in the state budget for higher education. To offset the reduction, the largest state university proposed a 25% tuition increase. It was determined that such an increase was needed simply to compensate for the lost support from the state. Separate random samples of 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors from the university were asked whether they were strongly opposed to the increase, given that it was the minimum increase necessary to maintain the university’s budget at current levels. The results are given in the table. Which null hypothesis would be appropriate for performing a chi-square test?

• The mean number of students who are strongly opposed is the same for each of the 4 years.

• The distribution of student opinion about the proposed tuition increase is the same for each of the 4 years at this university.

• There is an association between year in school and opinion about the tuition increase at this university.

• Year in school and student opinion about the tuition increase are independent in the sample.

• The closer students get to graduation, the less likely they are to be opposed to tuition increases.

Answer explanation

The null hypothesis should contain a statement that there is no difference in the true distribution of a categorical variable two 2 or more populations or no association between two categorical variables in a single population.

Recent revenue shortfalls in a midwestern state led to a reduction in the state budget for higher education. To offset the reduction, the largest state university proposed a 25% tuition increase. It was determined that such an increase was needed simply to compensate for the lost support from the state. Separate random samples of 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors from the university were asked whether they were strongly opposed to the increase, given that it was the minimum increase necessary to maintain the university’s budget at current levels. The results are given in the table. The conditions for carrying out chi-square test for homogeneity are:

I. Independent random samples from the populations of interest.

II. All sample sizes are less than 10% of the populations of interest.

III. All expected counts are at least 5.

Which of the conditions is (are) satisfied in this case?

• II only

• I only

• II and III only

• I and III only

• I, II, and III

Answer explanation

Did you check the expected counts and notice that the sampling was done without replacement?

A random sample of traffic tickets given to motorists in a large city is examined. The tickets are classified according to the race or ethnicity of the driver.The results are summarized in the table. The proportion of this city’s population in each of the racial/ethnic categories is listed in the table. We wish to test 𝐻₀: The racial/ethnic distribution of traffic tickets in the city is the same as the racial/ethnic distribution of the city’s population.

Assuming 𝐻₀ is true, what is the expected number of Hispanic drivers who would receive a ticket?

• 8

• 11.84

• 10.36

• 12

• 11

Answer explanation

8% of the population is Hispanic and there was a sample size of 148, so the expected count is 148(0.08).

Tags

CCSS.HSS.MD.A.2

A random sample of traffic tickets given to motorists in a large city is examined. The tickets are classified according to the race or ethnicity of the driver.The results are summarized in the table. The proportion of this city’s population in each of the racial/ethnic categories is listed in the table. We wish to test 𝐻₀: The racial/ethnic distribution of traffic tickets in the city is the same as the racial/ethnic distribution of the city’s population.

We compute the value of the 𝜒2 test statistic to be 6.57. Assuming that the conditions for inference are met, which of the following is the correct 𝑃-value?

• Between 0.05 and 0.10

• Between 0.10 and 0.20

• Greater than 0.20

• Less than 0.01

• Between 0.01 and 0.05

Answer explanation

With df = 3, the P-value is 0.087. Table C to find the 𝑃-value. Make sure that you are using the correct degrees of freedom: (4−1).

A random sample of traffic tickets given to motorists in a large city is examined. The tickets are classified according to the race or ethnicity of the driver.The results are summarized in the table. The proportion of this city’s population in each of the racial/ethnic categories is listed in the table. We wish to test 𝐻₀: The racial/ethnic distribution of traffic tickets in the city is the same as the racial/ethnic distribution of the city’s population.

The category that contributes the largest component to the 𝜒2 test statistic is

• White, with 12.4 more tickets than expected.

• White, with 12.4 fewer tickets than expected.

• Hispanic, with 6.16 more tickets than expected.

• Hispanic, with 6.16 fewer tickets than expected.

• Other, with 1.36 fewer tickets than expected.

Which of the following statements about chi-square distributions are true?

I. For all chi-square distributions, 𝑃(𝜒2≥0)=1.

II. A chi-square distribution with fewer than 10 degrees of freedom is roughly symmetric.

III. The more degrees of freedom a chi-square distribution has, the larger the mean of the distribution.

• I, II, and III

• III only

• I and III

• I only

• II only

Answer explanation

Chi-square distributions only take on positive values, so is true. A chi-square distribution with fewer than 10 df is skewed right. It is also true that as the df increases, so does the mean of the distribution.

Which of the following is 𝑛𝑜𝑡 one of the conditions that must be satisfied in order to perform inference about the slope of a least-squares regression line?

• The standard deviation 𝜎 of the population of 𝑦-values corresponding to a given value of 𝑥 is always the same, regardless of the specific value of 𝑥.

• The data come from a random sample or a randomized experiment.

• The sample size—that is, the number of paired observations (𝑥,𝑦)—exceeds 30.

• For each value of 𝑥, the population of 𝑦-values is Normally distributed.

• There exists a straight line such that, for each value of 𝑥, the mean 𝜇𝑦 of the corresponding population of 𝑦-values lies on that straight line.

Answer explanation

There is no sample size limitation in regression.

Tags

CCSS.HSS.ID.B.6B

Inference about the slope 𝛽 of a least-squares regression line is based on which of the following distributions?

• The standard Normal distribution

• The 𝑡 distribution with 𝑛 – 1 degrees of freedom

• The 𝑡 distribution with 𝑛 – 2 degrees of freedom

• The Normal distribution with mean 𝜇 and standard deviation 𝜎

• The chi-square distribution with 𝑛 – 1 degrees of freedom

Answer explanation

There are two variables, so the degrees of freedom is 𝑛 – 2.

Tags

CCSS.HSS.ID.A.4

An old saying in golf is “You drive for show and you putt for dough.” The point is that good putting is more important than long driving for shooting low scores and hence winning money. To see if this is the case, data from a random sample of 69 of the nearly 1000 players on the PGA Tour’s world money list are examined. The average number of putts per hole (fewer is better) and the player’s total winnings for the previous season are recorded and a least-squares regression line was fitted to the data. Assume the conditions for inference about the slope are met. Given is computer output from the regression analysis. Suppose that the researchers test the hypotheses 𝐻0:𝛽=0 versus 𝐻𝑎:𝛽<0 Which of the following is the value of the 𝑡 statistic for this test?

• –20.24

• 2.61

• – 2.44

• 0.081

• 2.44

Answer explanation

An old saying in golf is “You drive for show and you putt for dough.” The point is that good putting is more important than long driving for shooting low scores and hence winning money. To see if this is the case, data from a random sample of 69 of the nearly 1000 players on the PGA Tour’s world money list are examined. The average number of putts per hole (fewer is better) and the player’s total winnings for the previous season are recorded and a least-squares regression line was fitted to the data. Assume the conditions for inference about the slope are met. Given is computer output from the regression analysis. The 𝑃-value is 0.0087 for the test of the hypotheses 𝐻0:𝛽=0 versus 𝐻𝑎:𝛽<0. Which of the following is a correct interpretation of this result?

• If there is no linear relationship between average number of putts per hole and total winnings for the players on the PGA Tour’s world money list, the probability of getting a random sample of 69 players that yields a least-squares regression line with a slope of –4,139,198 or less is 0.0087.

• If there is no linear relationship between average number of putts per hole and total winnings for the players in the sample, the probability of getting a random sample of 69 players that yields a least-squares regression line with a slope of –4,139,198 or less is 0.0087.

• The probability there is no linear relationship between average number of putts per hole and total winnings for these 69 players is 0.0087.

• The probability there is no linear relationship between average number of putts per hole and total winnings for all players on the PGA Tour’s world money list is 0.0087.

• The probability of making a Type I error is 0.0087.

Answer explanation

Remember that the 𝑃-value measures how likely it is to get a sample result at least as extreme as the observed result, assuming that the null hypothesis is true from the population.