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 provided below.

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)? Narrower
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. In repeated sampling, 95% of the intervals constructed would contain the population mean.
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. 5.28 plus/minus 1.86(0.4/3) = (5.03, 5.53)
2. The consumer watchdog group claims that the mean time to respond exceeds 5 days and their stated reliability is 95%. Do you agree? Yes, since t = (5.28-5)/(0.4/3) = 2.10 is greater than 1.860.
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. NH: E(Y) = 2 vs. AH: E(Y) > 2
2. Calculate the test statistic for this test. t = (2.3-2)/(0.6/4) = 2.0
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.) The professors, since t-statistic=2 > 95th percentile=1.78
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.) 5th percentile=-1.66 < t-statistic=-1.47 < 10th percentile=-1.29, so 0.05< p/2 < 0.10, so 0.10 < p < 0.20
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. $280,000
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.) t-statistic < -2.05 or > +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%? Reject the null hypothesis since t-statistic=2.26 > 2.05 (or equivalently, since two-tail p-value=0.03 < significance level=0.05)
In a hypothesis test, is it true that the smaller the p-value, the less likely you are to reject the null hypothesis? No

Last updated: May, 2008

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