# Two-Sample Tests

I would like the response to the questions below in excel format, so that I can see the formulas used.

10.11

Digital cameras have taken over the majority of the point-and-shoot camera market. One of the important features of a camera is the battery life, as measured by the number of shots taken unit the battery needs to be recharged. The file DigitalCameras contains the battery life of 29 sub-compact cameras and 16 compact cameras (data extracted from "Digital Cameras," Consumer Reports, July 2009, pp. 28-29).

Battery Life Camera Type Subcompact Compact

320 Subcompact 320 520

520 Subcompact 520 260

160 Subcompact 160 400

160 Subcompact 160 200

300 Subcompact 300 300

120 Subcompact 120 150

520 Subcompact 520 360

440 Subcompact 440 200

300 Subcompact 300 260

170 Subcompact 170 80

150 Subcompact 150 200

300 Subcompact 300 260

180 Subcompact 180 400

100 Subcompact 100 260

150 Subcompact 150 200

400 Subcompact 400

170 Subcompact 170

180 Subcompact 180

160 Subcompact 160

240 Subcompact 240

320 Subcompact 320

260 Subcompact 260

150 Subcompact 150

320 Subcompact 320

180 Subcompact 180

140 Subcompact 140

220 Subcompact 220

160 Subcompact 160

340 Subcompact 340

200 Subcompact 200

130 Subcompact 130

520 Compact

260 Compact

400 Compact

200 Compact

300 Compact

150 Compact

360 Compact

200 Compact

260 Compact

80 Compact

200 Compact

260 Compact

400 Compact

260 Compact

200 Compact

a. Assuming that the population variances from both types of digital cameras are equal, is there evidence of a difference in the mean battery life between the two types of digital cameras (α=0.05)?

Determine the p-value in (a) and interpret its meaning.

b. Determine the p-value in (a) and interpret its meaning

c. Assuming that the population variances from both types of digital cameras are equal, construct and interpret a 95% confidence interval estimate of the difference between the population mean battery life of the two types of digital cameras.

10.21

In industrial settings, alternative methods often exist for measuring variables of interest. The data in the file (coded to maintain confidentiality) represent measurements in-line that were collected from an analyzer during the production process and from an analytical lab (extracted from M. Leitnaker, "Comparing Measurement Processes: In-line Versus Analytical

Measurements," Quality Engineering, 13, 2000-2001, pp. 293-298).

Sample In-Line Analytical lab

1 8.01 8.01

2 7.56 7.29

3 7.47 7.54

4 7.4 7.42

5 7.83 7.8

6 7.5 7.65

7 6.86 6.93

8 7.31 7.46

9 7.45 7.6

10 7.23 7.4

11 7.37 7.5

12 7.49 7.41

13 6.21 6.25

14 6.68 6.54

15 5.12 5.2

16 4.84 4.7

17 4.84 4.82

18 5.21 5.33

19 5.35 5.3

20 5.6 5.4

21 5.32 5.39

22 5.16 5.17

23 5.66 5.5

24 6.31 6.24

a. At the 0.05 level of significance, is there evidence of a difference in the mean measurements in-line and from and analytical lab?

b. What assumption is necessary about the population distribution in order to perform this test?

c. Use a graphical method to evaluate the validity of the assumption in (a).

d. Construct and interpret a 95% confidence interval estimate of the difference in the mean measurements in-line and from an analytical lab.

10.23

In tough economic times, magazines and other media have trouble selling advertisements. Thus, one indicator of a weak economy is a reduction in the number of magazine pages devoted to advertisements. The file Ad Pages (attached) contains the number of pages devoted to advertisements in May 2008 and May 2009 for 12 men's magazines.

a. At the 0.05 level of significance, is there evidence that the mean number of pages devoted to advertisements in men's magazines was higher in May 2008 than in May 2009?

b. What assumption is necessary about the population distribution in order to perform this?

c. Use a graphical method to evaluate the validity of the assumption in (b).

d. Construct and interpret a 95% confidence interval estimate of the difference in the mean number of pages devoted to advertisements in men's magazines between May 2008 and May 2009.

https://brainmass.com/statistics/descriptive-statistics/two-sample-tests-548032

#### Solution Summary

Two-sample tests for digital camera majorities are determined. Population variances for both types of digital cameras are given.

ANOVA Problem

When we want to test two samples to determine if it is likely that the population means (estimated by the sample means) are different, we typically use a t-test. If the samples are large, we can also use a z-test. (Note that the formulas for computing s, t and/or z in the case of a two-sample test are different than the formulas for computing the same values in a one-sample test. Use Excel data analysis to conduct tests comparing two sample means.)

Using ANOVA (short for Analysis of Variance), however, we can test 3 or more sample means to determine if at least one of the sample means comes from a population with a mean that is significantly different from all of the others in the test. We actually do this by estimating a combined population variance two different ways and comparing the two estimates (the ratio of these two variance estimates follows the so-called "F distribution").

Question:

Why do we need a new test method to compare the means of 3 or more populations? Why can't we just use a series of z-tests or t-tests to compare all of the possible pairs of population means to see if one (or more) is different?

Most of the testing is to determine one or two things:

1. Is there a statistically significant difference between two or more population means? (based on comparison of 2 or more sample means)

2. Is there a statistically significant relationship between two or more variables? We can use regression analysis or chi-square tests to answer this second question.)

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