## Sample Tests for a Population Mean

This section talks about using the central limit theorem to test a population mean when the sample size is large. It also addresses how to interpret the test results in the application background. Then, it discusses testing a population mean when the sample size is small, outlines a five-step testing procedure, and illustrates the procedure with an example. Study the example carefully and complete the relevant exercises and applications. Finally, it talks about large sample tests for a population proportion. The critical value and p-value approach are introduced based on a standardized test statistic.

### Large Sample Tests for a Population Proportion

#### BASIC

On all exercises for this section you may assume that the sample is sufficiently large for the relevant test to be validly performed.

1. Compute the value of the test statistic for each test using the information given.

1. Testing $H_{0}: p=0.50$ vs. $H_{a}: p 0.50, n=360, \hat{p}=0.56$.
2. Testing $H_{0}: p=0.50$ vs. $H_{a}: p \neq 0.50, n=360, \hat{p}=0.56$.
3. Testing $H_{0}: p=0.37$ vs. $H_{a}: p < 0.37, n=1200, \hat{p}=0.35$.

3. For each part of Exercise 1 construct the rejection region for the test for $\alpha=0.05$ and make the decision based on your answer to that part of the exercise.

5. For each part of Exercise 1 compute the observed significance $(p$-value) of the test and compare it to $\alpha=0.05$ in order to make the decision by the $p$-value approach to hypothesis testing.

7. Perform the indicated test of hypotheses using the critical value approach.

1. Testing $H_{0}: p=0.55$ vs. $H_{a}: p > 0.55 @ \alpha=0.05, n=300, \hat{p}=0.60$.
2. Testing $H_{0}: p=0.47$ vs. $H_{a}: p \neq 0.47 @ \alpha=0.01, n=9750, \hat{p}=0.46$.

9. Perform the indicated test of hypotheses using the $p$-value approach.

1. Testing $H_{0}: p=0.37$ vs. $H_{a}: p \neq 0.37 @ \alpha=0.005, n=1300, \hat{p}=0.40$.
2. Testing $H_{0}: p=0.94$ vs. $H_{a}: p > 0.94 @ \alpha=0.05, n=1200, \hat{p}=0.96$.

#### APPLICATIONS

11. Five years ago 3.9% of children in a certain region lived with someone other than a parent. A sociologist wishes to test whether the current proportion is different. Perform the relevant test at the 5% level of significance using the following data: in a random sample of 2,759 children, 119 lived with someone other than a parent.

13. Two years ago 72% of household in a certain county regularly participated in recycling household waste. The county government wishes to investigate whether that proportion has increased after an intensive campaign promoting recycling. In a survey of 900 households, 674 regularly participate in recycling. Perform the relevant test at the 10% level of significance.

15. A report five years ago stated that 35.5% of all state-owned bridges in a particular state were "deficient". An advocacy group took a random sample of 100 state-owned bridges in the state and found 33 to be currently rated as being "deficient". Test whether the current proportion of bridges in such condition is 35.5% versus the alternative that it is different from 35.5%, at the 10% level of significance.

17. According to the Federal Poverty Measure 12% of the U.S. population lives in poverty. The governor of a certain state believes that the proportion there is lower. In a sample of size 1,550, 163 were impoverished according to the federal measure.

1. Test whether the true proportion of the state's population that is impoverished is less than 12%, at the 5% level of significance.
2. Compute the observed significance of the test.

19. A special interest group asserts that 90% of all smokers began smoking before age 18. In a sample of 850 smokers, 687 began smoking before age 18.

1. Test whether the true proportion of all smokers who began smoking before age 18 is less than 90%, at the 1% level of significance.
2. Compute the observed significance of the test.

21. A rule of thumb is that for working individuals one-quarter of household income should be spent on housing. A financial advisor believes that the average proportion of income spent on housing is more than 0.25. In a sample of 30 households, the mean proportion of household income spent on housing was 0.285 with a standard deviation of 0.063. Perform the relevant test of hypotheses at the 1% level of significance. Hint: This exercise could have been presented in an earlier section.

#### LARGE DATA SET EXERCISES

23. Large Data Sets 4 and 4A list the results of 500 tosses of a die. Let p denote the proportion of all tosses of this die that would result in a five. Use the sample data to test the hypothesis that p is different from 1/6, at the 20% level of significance.

http://www.gone.2012books.lardbucket.org/sites/all/files/data4.xls

http://www.gone.2012books.lardbucket.org/sites/all/files/data4A.xls

25. Lines 2 through 536 in Large Data Set 11 is a sample of 535 real estate sales in a certain region in 2008. Those that were foreclosure sales are identified with a 1 in the second column. Use these data to test, at the 10% level of significance, the hypothesis that the proportion p of all real estate sales in this region in 2008 that were foreclosure sales was less than 25%. (The null hypothesis is $H_{0}: p=0.25$).

http://www.gone.2012books.lardbucket.org/sites/all/files/data11.xls