# Analysis of variance and confidence interval, chi-square

Out of the 15 study problems I am having trouble with 6. I want to compare my answers to yours. Can you please work the following problems, explaing how you got the answer and showing the tables from excel, please. It is very important I understand how you got the answers.

1. A sample of the reading scores of 25 sixth-graders has a mean of 82, and the standard deviation of the sample is 15. Assume that the population can be adequately approximated by a normal distribution.

(a) Find the 95% confidence interval of the mean reading scores of all sixth-graders.

(b) Find the 85% confidence interval of the mean reading scores of all six-graders

2. The times (in minutes) it took six white mice to learn to run a simple maze and the times it took six brown mice to learn to run the same maze are given here.

White Mice Brown Mice

18 25

24 16

20 19

13 14

15 16

12 10

(a) Does the color of the mice make a difference in their learning rate? Test using a significance level of 5%.

(b) Give the p-value for the test, and interpret this value.

(c) Find the 99% confidence interval for the difference of the means. Interpret this interval.

Note: You can assume that the data is normally distributed.

3. Three different relaxation techniques are given to randomly selected patients in an effort to reduce their stress levels. A special instrument has been designed to measure the percentage of stress reduction in each person. The data is shown below. You can assume normality and that good randomization and experimental procedures were used.

Relaxation Experiment

Technique I Technique II Technique III

3 12 15

10 12 14

5 17 18

1 13 14

13 18 20

3 9 22

4 14 16

Carry out a "complete" analysis using ANOVA and a 5% significance level.

4. A researcher wishes to determine if the number of hours (Y) a person exercises per week is related to their age (X). The data is shown below, and you can assume a well designed experiment was used to obtain the data.

Age (X): 18 22 26 32 35 38 52 59

Hours (Y): 10 8 5 2 4 3 1.5 1

Carry out a "complete" analysis using Simple Linear Regression and a 5% significance level. In addition, use your results to estimate the average amount of exercise for a 29 year old person, and provide a 98% C.I. for your estimate. Remember in your analysis to discuss the pertinent results obtained.

5. (a) A researcher surveyed married women and single women to ascertain whether there was a difference in the number of books each had read during the past year. The data is shown below.

Books Read

Married Single

6 2

8 3

7 5

4 11

9 3

12 5

13 11

7 12

10 16

18 4

15 0

1

You can't assume normality, so use an appropriate nonparametric method to

Test the claim that each group read the same number of books. Test using a

significance level of 10%. Draw appropriate conclusions.

(b) You are curious what the result would be if you used a parametric technique,

so please re-do the analysis using the appropriate two sample t-test. Comment on

the result.

6. An experiment was conducted to study the effects of temperature of freezer (X1) and freezer storage density (X2) on number of days before flavor deterioration (Y) occurs, for a food product stored in a commercial freezer. The independent variables are measured in terms of deviations from the levels normally used; thus in the first observation the temperature setting was 10 degrees centigrade below the normal setting and the storage setting was 10 percentage points less than the normal density. Note: This coding should not concern you, and has nothing to do with carrying out the regression. The results of the study are as follows:

Y: 196 169 138 179 158 122 164 139 108

X1: -10 0 +10 -10 0 +10 -10 0 +10

X2: -10 -10 -10 0 0 0 +10 +10 +10

The following model is proposed:

Y = b0 + b1x1 + b2x2 + e

Carry out a "complete" multiple regression analysis of this model. In addition predict y for x1 = -5, and x2 = 5, and give a 95% C.I. for both Y|X, and E[Y|X].

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#### Solution Preview

Kathy Scott

Study Guide Questions (6 of 15)

Final Exam Study Guide

Answers in RED

1. A sample of the reading scores of 25 sixth-graders has a mean of 82, and the standard deviation of the sample is 15. Assume that the population can be adequately approximated by a normal distribution.

(a) Find the 95% confidence interval of the mean reading scores of all sixth-graders.

where is the t- critical value at (n-1) deg. of freedom. We use t-distribution because we estimate the standard deviation.

= (78.8, 89.2).

Always interpret: For example,

We are 95% confident that the pop. Mean test score lies between 78.8 and 89.2.

(b) Find the 85% confidence interval of the mean reading scores of all six-graders

Same as above, the difference is just the critical value. Here, . Now, just replace 2.39 above with 1.86 and show that the interval is (76.4, 87.6). Interpret.

2. The times (in minutes) it took six white mice to learn to run a simple maze and the times it took six brown mice to learn to run the same maze are given here.

White Mice Brown Mice

18 25

24 16

20 19

13 14

15 16

12 10

Xbar1 = 17 xbar2 = 16.67

Stdev1 =4.56 stdev2 = 5.05

Var1 = 20.8 Var2 = 25.47

n1 = 6 n2 = 6

(a) Does the color of the mice make a difference in their learning rate? Test using a significance level of 5%.

I will test difference in mean of the two color mice.

Ho : (No difference between the mean time)

Ha : (Significant difference between the mean time)

Test stats = (17-16.67)/√ (20.8/6 + 25.47/6) = 0.119

T-critical = 2.64

Since 0.119 < 2.64, do not reject Ho and conclude there is no significant difference between the mean times of the two mice. Thus, color does not make a difference.

(b) Give the p-value for the test, and interpret this value.

Pvalue = P(T > 0.119) = 0.9068 (using Excel software).

P-value > 2(0.20) = 0.40 (using the t- table). Notice that I am using n-2 = 12-2 = 10 degree of freedom.

(c) Find the 99% confidence interval for the difference of the means. Interpret this interval.

Note: You can assume that the data is normally distributed.

Just like in question 1 above but with different values:

= (-9.61, 10.27)

We are 99% confident that the difference between the population means of the two mice ...

#### Solution Summary

Out of the 15 study problems I am having trouble with 6. I want to compare my answers to yours. Can you please work the following problems, explaing how you got the answer and showing the tables from excel, please. It is very important I understand how you got the answers.

1. A sample of the reading scores of 25 sixth-graders has a mean of 82, and the standard deviation of the sample is 15. Assume that the population can be adequately approximated by a normal distribution.

(a) Find the 95% confidence interval of the mean reading scores of all sixth-graders.

(b) Find the 85% confidence interval of the mean reading scores of all six-graders