AP Stats Chapter 11

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 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.

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.

Is this a probability distribution?

• No

• Yes

Is this a probability distribution?

• No, the sum of p(x) does not equal 1.

• Yes, all p(x) are between 0 and 1.

• No, all p(x) are not between 0 and 1.

• Yes, the sum p(x) is 1.

• A

• B

• C

• D

• E

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.