Applied Regression Modeling: A Business Approach

Problems on statistical foundations

The following problems enable students to practice important concepts from Chapter 1, without having to use a computer. Answers are available here.

In the construction of confidence intervals, will an increase in the sample size lead to a wider or narrower interval (if all other quantities are unchanged)?
Suppose a 95% confidence interval for the population mean, E(Y), turns out to be (50, 105). Give a definition of what it means to be "95% confident" here.
A consumer watchdog group has concerns about the length of time a company takes to respond to complaints about its products. Studies show a mean time to respond of 5.28 days and standard deviation of 0.4 days for a sample of n = 9 complaints. (You may find the following information useful in answering the subsequent questions: the 90th percentile of the t-distribution with 8 degrees of freedom is 1.397; the 95th percentile of the t-distribution with 8 degrees of freedom is 1.860.)
1. Construct a 90% confidence interval for the mean time to respond.
2. The consumer watchdog group claims that the mean time to respond exceeds 5 days and their stated reliability is 95%. Do you agree?
Students have claimed that the average number of classes missed per student during a quarter is 2. College professors dispute this claim and believe the average is more than this. They sample n = 16 students and find the sample mean is 2.3 and the sample standard deviation is 0.6.
1. State the null and alternative hypothesis that the professors wish to test.
2. Calculate the test statistic for this test.
3. Using a 5% significance level, who appears to be correct, the students or the professors? (You may find some of the following information useful: the 95th percentile of the t-distribution with 12 degrees of freedom is 1.78; the 97.5th percentile of the t-distribution with 12 degrees of freedom is 2.18.)
Consider the following computer output:
```
	--------------------------------------------------------
	NULL HYPOTHESIS:  Pop. MEAN of Y = 3
		
	Y			=	Sales

	SAMPLE MEAN OF Y	=	2.97
	SAMPLE VARIANCE OF Y	=	0.25
	SAMPLE SIZE OF Y	=	150
	
	t-statistic		=	-1.47
	--------------------------------------------------------
```
Suppose a two-tailed test for the alternative hypothesis that the population mean is not equal to 3 is desired. Find upper and lower limits for the p-value for the test. (You may find some of the following information useful: the 90th percentile of the t-distribution with 149 degrees of freedom is 1.29; the 95th percentile of the t-distribution with 166 degrees of freedom is 1.66.)
A realtor would like to see if the average sale price of the homes in a particular neighborhood changed in the last 12 months. A study conducted 12 months ago indicated that the average sale price of neighborhood homes was $280,000. Data was collected and the following computer output generated:
```
	---------------------------------------------------
	NULL HYPOTHESIS:  Pop. MEAN of Y = ?
		
	Y			=	Sale_Price

	SAMPLE MEAN OF Y	=	289,280
	SAMPLE SIZE OF Y	=	28
		
	t-statistic		=	2.26
	TWO-TAILED P-VALUE	=	0.03
	---------------------------------------------------
```
1. Identify the missing value in the null hypothesis statement in this problem.
2. Specify the rejection region for conducting a two-tailed test at significance level 5%. (You may find some of the following information useful: the 95th percentile of the t-distribution with 27 degrees of freedom is 1.70; the 97.5th percentile of the t-distribution with 27 degrees of freedom is 2.05.)
3. Based on the information presented in this printout, would you reject or fail to reject the null hypothesis if conducting a two-tailed test at significance level 5%?
In a hypothesis test, is it true that the smaller the p-value, the less likely you are to reject the null hypothesis?

Last updated: May, 2008

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