Inference Test AP Stats

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.

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.

A 95% confidence interval for µ based on 𝑛 = 15 observations from a Normal population is (–0.73, 1.92). If we use this confidence interval to test the hypothesis 𝐻₀: 𝜇 = 0 against 𝐻_a: 𝜇 ≠ 0, what is the appropriate conclusion?

​ (a)   H₀ at the α = ​ (b)  

• fail to reject

• reject

• .05

• .10

Answer explanation

Because the hypothesized value, 0, is one of the plausible values in the 95% confidence interval, we fail to reject 𝐻₀:𝜇 = 0 at the α = 0.05 level.

You are thinking of conducting a one-sample 𝑡 test about a population mean 𝜇 using a 0.05 significance level. Which of the following statements is correct?

• You can carry out the test only if the population standard deviation is known.

• You can safely carry out the test if there are no outliers, regardless of the sample size.

• You can safely carry out the test if your sample size is at least 30.

• You can carry out the test if a graph of the data shows no strong skewness, regardless of the sample size.

• You should not carry out the test if the sample does not have a Normal distribution.

Answer explanation

We cannot use 𝑡 procedures if the graph of the sample data shows strong skewness OR outliers.

Which of the following has the smallest probability?

• 𝑃(𝑧>2) if 𝑧 is a standard Normal random variable.

• 𝑃(𝑡>2)if 𝑡 has 2 degrees of freedom.

• 𝑃(𝑧<2) if 𝑧 is a standard Normal random variable.

• 𝑃(𝑡<2) if 𝑡 has 5 degrees of freedom.

• 𝑃(𝑡>2)if 𝑡 has 5 degrees of freedom.

Answer explanation

The standard Normal distribution has less area in the tail than a 𝑡 distribution with df = 2 or df = 5.

A significance test was performed to test 𝐻₀ : 𝜇 = 2 versus the alternative 𝐻_a: 𝜇 ≠ 2. A sample of size 28 produced a standardized test statistic of 𝑡 = 2.051. Assume all conditions for inference are met. Using technology the 𝑃-value is  (Round to 4 decimal places)   ___?

Answer explanation

tcdf(2.051, 1000, 27)

Double the answer to account for both tails of the distribution.

Using Table B and df = 27, 𝑡 = 2.051 falls between 0.025 and 0.05. Because this is a two-sided test, the 𝑃-value falls between 0.05 and 0.10.

A study of road rage asked separate random samples of 596 men and 523 women about their behavior while driving. Based on their answers, each respondent was assigned a road rage score on a scale of 0 to 20. Are the conditions for performing a two-sample 𝑡 test satisfied?

• Maybe; we have independent random samples, but we should look at the data to check Normality.

• Yes; the large sample sizes guarantee that the corresponding population distributions will be Normal.

• No; we don't know the population standard deviations.

• No; road rage scores on a scale from 0 to 20 can't be Normal.

• Yes; we have two independent random samples and large sample sizes.

Answer explanation

Because the values are on a scale of 1 to 20, the populations can’t be Normal. However, the large sample sizes justify the use of the two-sample 𝑡 test. Also, the data come from independent random samples that are both less than 10% of their respective populations.

A researcher wished to compare the average amount of time spent in extracurricular activities by high school students in a suburban school district with that in a school district of a large city. The researcher obtained an SRS of 60 high school students in a large suburban school district and found the mean time spent in extracurricular activities per week to be 6 hours with a standard deviation of 3 hours. The researcher also obtained an independent SRS of 40 high school students in a large city school district and found the mean time spent in extracurricular activities per week to be 5 hours with a standard deviation of 2 hours. Suppose that the researcher decides to carry out a significance test of 𝐻₀:𝜇𝑠𝑢𝑏𝑢𝑟𝑏𝑎𝑛=𝜇𝑐𝑖𝑡𝑦 versus a two-sided alternative.

Which is the correct standardized test statistic?

• 
• 
• 
• 
• 

Answer explanation

A test for the difference between two means with population standard deviations unknown is a 𝑡 test.

A researcher wished to compare the average amount of time spent in extracurricular activities by high school students in a suburban school district with that in a school district of a large city. The researcher obtained an SRS of 60 high school students in a large suburban school district and found the mean time spent in extracurricular activities per week to be 6 hours with a standard deviation of 3 hours. The researcher also obtained an independent SRS of 40 high school students in a large city school district and found the mean time spent in extracurricular activities per week to be 5 hours with a standard deviation of 2 hours. Suppose that the researcher decides to carry out a significance test of 𝐻0:𝜇suburban = 𝜇city versus a two-sided alternative. The 𝑃-value for the test is 0.048. A correct conclusion is to

• fail to reject 𝐻0 because 0.048<𝛼=0.050. There is convincing evidence that the average time spent on extracurricular activities by students in the suburban and city school districts is the same.

• fail to reject 𝐻0 because 0.048<𝛼=0.050. There is not convincing evidence of a difference in the average time spent on extracurricular activities by students in the suburban and city school districts.

• fail to reject 𝐻0 because 0.048<𝛼=0.050. There is convincing evidence of a difference in the average time spent on extracurricular activities by students in the suburban and city school districts.

• reject 𝐻0 because 0.048<𝛼=0.050. There is convincing evidence of a difference in the average time spent on extracurricular activities by students in the suburban and city school districts.

• reject 𝐻0 because 0.048<𝛼=0.050. There is not convincing evidence of a difference in the average time spent on extracurricular activities by students in the suburban and city school districts.

Answer explanation

With the 𝑃-value less than 0.05, we reject 𝐻₀ and have convincing evidence for the alternative hypothesis.

Are TV commercials louder than their surrounding programs? To find out, researchers collected data on 50 randomly selected commercials in a given week. With the television’s volume at a fixed setting, they measured the maximum loudness of each commercial and the maximum loudness in the first 30 seconds of regular programming that followed. Assuming conditions for inference are met, the most appropriate method for answering the question of interest is

• a two-sample 𝑧 test for a difference in proportions.

• a two-sample 𝑡 interval for a difference in means.

• a two-sample 𝑡 test for a difference in means.

• a paired 𝑡 interval for a mean difference.

• a paired 𝑡 test for a mean difference.

Answer explanation

For each commercial selected, there were two measurements being made. These are unlikely to be independent because they were on the same channel at nearly the same time.