# Inference about a Population Mean

## Presentation on theme: "Inference about a Population Mean"— Presentation transcript:

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 Changes to a t-distribution. Stays as the standardized Normal. Changes to an estimate of the Normal distribution.

When we replace  with s in our standardized test statistic formula, the distribution of the test statistic Changes to a t-distribution. Stays as the standardized Normal. Changes to an estimate of the Normal distribution.

t-distribution When do we use the t-distribution to make inference about ? When we know  but  is unknown. When we have a very large sample size. When the data are very skewed or when outliers are present. When we do not know  or .

When do we use the t-distribution to make inference about ? When we know  but  is unknown. When we have a very large sample size. When the data are very skewed or when outliers are present. When we do not know  or .

Standard error What is NOT true about the standard error of the sample mean, ? It estimates We need  when computing it. We use it when we do not know . It is an estimate for the standard deviation of

What is NOT true about the standard error of the sample mean, ? It estimates We need  when computing it. We use it when we do not know . It is an estimate for the standard deviation of

t-distribution Which of the following graphs could be a t-distribution? Plot A Plot B Plot C

Which of the following graphs could be a t-distribution? Plot A Plot B Plot C

t-distribution The following graph shows three different density curves. If Line B shows a t-distribution, which line could NOT show a normal distribution? Line A Line C

The following graph shows three different density curves. If Line B shows a t-distribution, which line could NOT show a normal distribution? Line A Line C

t-table 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? 1.645 2.306 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? 1.645 2.306 2.365

t-table 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? 1.645 1.671 1.676 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? 1.645 1.671 1.676 2.009

Hypothesis testing 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 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?

Hypothesis testing 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? 0.001 < P-value < < P-value < 0.005 0.005 < P-value < 0.01 Impossible to determine since no negative numbers on the table.

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? 0.001 < P-value < < P-value < 0.005 0.005 < P-value < 0.01 Impossible to determine since no negative numbers on the table.

Hypothesis testing 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 < P-value < , what decision should be made if testing at the  = 0.05 level? Reject H0 and conclude that the tires were not performing as claimed. Reject H0 and conclude that the mean tire life really is 50,000 miles. Do not reject H0 and conclude that the tires were not performing as claimed. Do not reject H0 and conclude that the mean tire life really is 50,000 miles.

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 < P-value < , what decision should be made if testing at the  = 0.05 level? Reject H0 and conclude that the tires were not performing as claimed. Reject H0 and conclude that the mean tire life really is 50,000 miles. Do not reject H0 and conclude that the tires were not performing as claimed. Do not reject H0 and conclude that the mean tire life really is 50,000 miles.

t procedures 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?. Yes, because we can assume that the time spent studying was normally distributed. Yes, because the 45 students is a big enough sample to invoke the CLT. No, because time spent studying is not a quantitative variable. 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?. Yes, because we can assume that the time spent studying was normally distributed. Yes, because the 45 students is a big enough sample to invoke the CLT. No, because time spent studying is not a quantitative variable. No, because there was no randomization in choosing the sample.

t procedures 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? No, because the data are not bell-shaped and the sample size is not large enough to use the CLT. 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? No, because the data are not bell-shaped and the sample size is not large enough to use the CLT. Yes, because he used a random method with which to collect his data.

t procedures 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? Yes, because data were from an experiment. Yes, because the sample size is large enough. No, because the data are skewed and have outliers. 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? Yes, because data were from an experiment. Yes, because the sample size is large enough. No, because the data are skewed and have outliers. No, because we did not repeat the sampling enough times.

t procedures 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? No, because the sample size is not large enough. 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? No, because the sample size is not large enough. Yes, because the data show no strong skewness or outliers.