Inference about a Population Mean

BPS chapter 17

© 2010 W.H. Freeman and Company

Distributions When we replace with s in our standardized test statistic formula, the

distribution of the test statistic

a) Changes to a t-distribution.

b) Stays as the standardized Normal.

c) Changes to an estimate of the Normal distribution.

Distributions (answer) When we replace with s in our standardized test statistic formula, the

distribution of the test statistic

a) Changes to a t-distribution.

b) Stays as the standardized Normal.

c) Changes to an estimate of the Normal distribution.

t-distributionWhen do we use the t-distribution to make inference about ?

a) When we know but is unknown.

b) When we have a very large sample size.

c) When the data are very skewed or when outliers are present.

d) When we do not know or .

t-distribution (answer)When do we use the t-distribution to make inference about ?

a) When we know but is unknown.

b) When we have a very large sample size.

c) When the data are very skewed or when outliers are present.

d) When we do not know or .

Standard errorWhat is NOT true about the standard error of the sample mean, ?

a) It estimates .

b) We need when computing it.

c) We use it when we do not know .

d) It is an estimate for the standard deviation of .

s

n

n

x

Standard error (answer)What is NOT true about the standard error of the sample mean, ?

a) It estimates .

b) We need when computing it.

c) We use it when we do not know .

d) It is an estimate for the standard deviation of .

s

n

n

x

t-distributionWhich of the following graphs could be a t-distribution?

a) Plot A

b) Plot B

c) Plot C

t-distribution (answer)Which of the following graphs could be a t-distribution?

a) Plot A

b) Plot B

c) Plot C

t-distributionThe following graph shows three different density curves. If Line B

shows a t-distribution, which line could NOT show a normal distribution?

a) Line A

b) Line C

t-distribution (answer)The following graph shows three different density curves. If Line B

shows a t-distribution, which line could NOT show a normal distribution?

a) Line A

b) Line C

t-tableMany people believe that playing chess can have a positive effect on

reading ability. A sample of n = 8 chess players was taken. After participating in a comprehensive chess program, each individual was given a post-test to gauge reading ability. If we want to create a 95% confidence interval for the true mean score, what is the correct t* for this problem?

a) 1.645

b) 2.306

c) 2.365

t-table (answer)Many people believe that playing chess can have a positive effect on

reading ability. A sample of n = 8 chess players was taken. After participating in a comprehensive chess program, each individual was given a post-test to gauge reading ability. If we want to create a 95% confidence interval for the true mean score, what is the correct t* for this problem?

a) 1.645

b) 2.306

c) 2.365

t-tableMany people believe that playing chess can have a positive effect on

reading ability. A sample of n = 53 chess players was taken. After participating in a comprehensive chess program, each individual was given a post-test to gauge reading ability. If we want to create a 90% confidence interval for the true mean score, what is the correct t* for this problem?

a) 1.645

b) 1.671

c) 1.676

d) 2.009

t-table (answer)Many people believe that playing chess can have a positive effect on

reading ability. A sample of n = 53 chess players was taken. After participating in a comprehensive chess program, each individual was given a post-test to gauge reading ability. If we want to create a 90% confidence interval for the true mean score, what is the correct t* for this problem?

a) 1.645

b) 1.671

c) 1.676

d) 2.009

Hypothesis testingA tire manufacturer claims that one particular type of tire will last at

least 50,000 miles. A group of angry customers does not believe this is so. They take a sample of 14 tires and want to test if the mean mileage of the tires is really less than 50,000. What set of hypotheses are they interested in testing?

a)

b)

c)

d)

Hypothesis testing (answer)A tire manufacturer claims that one particular type of tire will last at

least 50,000 miles. A group of angry customers does not believe this is so. They take a sample of 14 tires and want to test if the mean mileage of the tires is really less than 50,000. What set of hypotheses are they interested in testing?

a)

b)

c)

d)

Hypothesis testingA tire manufacturer claims that one particular type of tire will last at

least 50,000 miles. A group of angry customers does not believe this is so. They take a sample of 14 tires and want to test if the mean mileage of the tires is really less than 50,000. If the test statistic came out to be –3.006, what is the range for the corresponding P-value?

a) 0.001 < P-value < 0.0025

b) 0.0025 < P-value < 0.005

c) 0.005 < P-value < 0.01

d) Impossible to determine since no negative numbers on the table.

Hypothesis testing (answer)A tire manufacturer claims that one particular type of tire will last at

least 50,000 miles. A group of angry customers does not believe this is so. They take a sample of 14 tires and want to test if the mean mileage of the tires is really less than 50,000. If the test statistic came out to be –3.006, what is the range for the corresponding P-value?

a) 0.001 < P-value < 0.0025

b) 0.0025 < P-value < 0.005

c) 0.005 < P-value < 0.01

d) Impossible to determine since no negative numbers on the table.

Hypothesis testingA tire manufacturer claims that one particular type of tire will last at

least 50,000 miles. A group of angry customers does not believe this is so. They take a sample of 14 tires and want to test if the mean mileage of the tires is really less than 50,000. If 0.0100 < P-value < 0.0200, what decision should be made if testing at the = 0.05 level?

a) Reject H0 and conclude that the tires were not performing as claimed.

b) Reject H0 and conclude that the mean tire life really is 50,000 miles.

c) Do not reject H0 and conclude that the tires were not performing as claimed.

d) Do not reject H0 and conclude that the mean tire life really is 50,000 miles.

Hypothesis testing (answer)A tire manufacturer claims that one particular type of tire will last at

least 50,000 miles. A group of angry customers does not believe this is so. They take a sample of 14 tires and want to test if the mean mileage of the tires is really less than 50,000. If 0.0100 < P-value < 0.0200, what decision should be made if testing at the = 0.05 level?

a) Reject H0 and conclude that the tires were not performing as claimed.

b) Reject H0 and conclude that the mean tire life really is 50,000 miles.

c) Do not reject H0 and conclude that the tires were not performing as claimed.

d) Do not reject H0 and conclude that the mean tire life really is 50,000 miles.

t proceduresA student wanted to assess the average time spent studying for his

most recent exam taken in class. He asked the first 45 students who came to class how much time they spent and recorded the values. He then used this information to calculate a 95% confidence interval for the mean time spent by all students. Was this an appropriate use of the t procedure for a confidence interval?.

a) Yes, because we can assume that the time spent studying was normally distributed.

b) Yes, because the 45 students is a big enough sample to invoke the CLT.

c) No, because time spent studying is not a quantitative variable.

d) No, because there was no randomization in choosing the sample.

t procedures (answer)A student wanted to assess the average time spent studying for his

most recent exam taken in class. He asked the first 45 students who came to class how much time they spent and recorded the values. He then used this information to calculate a 95% confidence interval for the mean time spent by all students. Was this an appropriate use of the t procedure for a confidence interval?.

a) Yes, because we can assume that the time spent studying was normally distributed.

b) Yes, because the 45 students is a big enough sample to invoke the CLT.

c) No, because time spent studying is not a quantitative variable.

d) No, because there was no randomization in choosing the sample.

t proceduresThe same student from the previous question also wanted to test if the

mean score for all students taking the exam was equal to 80%. He took a simple random sample of size n = 18 students from his class. After taking his sample, he created the following histogram:

Should he continue his analysis and perform this test of hypothesis?

a) No, because the data are not bell-shaped and the sample size is not large enough to use the CLT.

b) Yes, because he used a random method with which to collect his data.

t procedures (answer)The same student from the previous question also wanted to test if the

mean score for all students taking the exam was equal to 80%. He took a simple random sample of size n = 18 students from his class. After taking his sample, he created the following histogram:

Should he continue his analysis and perform this test of hypothesis?

a) No, because the data are not bell-shaped and the sample size is not large enough to use the CLT.

b) Yes, because he used a random method with which to collect his data.

t proceduresThe following histogram represents the yearly advertising budgets (in

millions of dollars) of 21 randomly selected companies. A statistics student wants to create a confidence interval for the mean advertising budget of all companies. By looking at the histogram, is the use of a t procedure appropriate in this case?

a) Yes, because data were from an experiment.

b) Yes, because the sample size is large enough.

c) No, because the data are skewed and have outliers.

d) No, because we did not repeat the sampling enough times.

t procedures (answer)The following histogram represents the yearly advertising budgets (in

millions of dollars) of 21 randomly selected companies. A statistics student wants to create a confidence interval for the mean advertising budget of all companies. By looking at the histogram, is the use of a t procedure appropriate in this case?

a) Yes, because data were from an experiment.

b) Yes, because the sample size is large enough.

c) No, because the data are skewed and have outliers.

d) No, because we did not repeat the sampling enough times.

t proceduresA train operator claims that her trains are never more than 7 minutes

late. A commuter takes a simple random sample of trains arriving at her local station of size n = 21 and records their estimated arrival time and the actual arrival time. She creates the following stemplot for these data:

Based on the stemplot, should she continue with her use of the t-procedure?

a) No, because the sample size is not large enough.

b) Yes, because the data show no strong skewness or outliers.

t procedures (answer)A train operator claims that her trains are never more than 7 minutes

late. A commuter takes a simple random sample of trains arriving at her local station of size n = 21 and records their estimated arrival time and the actual arrival time. She creates the following stemplot for these data:

Based on the stemplot, should she continue with her use of the t-procedure?

a) No, because the sample size is not large enough.

b) Yes, because the data show no strong skewness or outliers.