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G E R Are senior citizens who live in retirement communities more socially active than those who live in age-integrated communities? Write a sentence or two explaining the results of these tests. (HINT: Remember to use the proper formulas for small sample sizes.) a. A random sample of senior citizens living in a retirement village reported that they had an average of 1.42 face-to-face interactions per day with their neighbors. A random sample of those living in age-integrated communities reported 1.58 interactions. Is the difference significant? Sample 1 (Retirement Community) Sample 2 (Age-Integrated Neighborhood) X ¯ 1 = 1.42 X ¯ 2 = 1.58 s 1 = 0.10 s 2 = 0.78 N 1 = 43 N 2 = 37 b. Senior citizens living in the retirement village reported that they had an average of 7.43 telephone calls with friends and relatives each week, while those in the age-integrated communities reported an average of 5.50 calls. Is the difference significant? Sample 1 (Retirement Community) Sample 2 (Age-Integrated Neighborhood) X ¯ 1 = 7.43 X ¯ 2 = 5.50 s 1 = 0.75 s 2 = 0.25 N 1 = 43 N 2 = 37

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Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836
BuyFind

Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836

Solutions

Chapter
Section
Chapter 8, Problem 8.7P
Textbook Problem
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G E R Are senior citizens who live in retirement communities more socially active than those who live in age-integrated communities? Write a sentence or two explaining the results of these tests. (HINT: Remember to use the proper formulas for small sample sizes.)

a. A random sample of senior citizens living in a retirement village reported that they had an average of 1.42 face-to-face interactions per day with their neighbors. A random sample of those living in age-integrated communities reported 1.58 interactions. Is the difference significant?

Sample 1
(Retirement Community)
Sample 2
(Age-Integrated Neighborhood)
X ¯ 1 = 1.42 X ¯ 2 = 1.58
s 1 = 0.10 s 2 = 0.78
N 1 = 43 N 2 = 37

b. Senior citizens living in the retirement village reported that they had an average of 7.43 telephone calls with friends and relatives each week, while those in the age-integrated communities reported an average of 5.50 calls. Is the difference significant?

Sample 1
(Retirement Community)
Sample 2
(Age-Integrated Neighborhood)
X ¯ 1 = 7.43 X ¯ 2 = 5.50
s 1 = 0.75 s 2 = 0.25
N 1 = 43 N 2 = 37

Expert Solution
To determine

(a)

To find:

If the senior citizens who live in retirement communities are more socially active than those who live in age integrated communities.

Explanation of Solution

Given:

The sample statistics is given in the table below,

Sample 1
(Retirement Community)
Sample 2
(Age-Integrated Neighborhood)
X¯1=1.42 X¯2=1.58
s1=0.10 s2=0.78
N1=43 N2=37

Approach:

The five step model for hypothesis testing is,

Step 1. Making assumptions and meeting test requirements.

Step 2. Stating the null hypothesis.

Step 3. Selecting the sampling distribution and establishing the critical region.

Step 4. Computing test statistics.

Step 5. Making a decision and interpreting the results of the test.

Formula used:

The formula to calculate the sampling distribution of the differences in sample means of small samples is given by,

t(obtained)=(X¯1X¯2)σX¯X¯

Where, X¯1 and X¯2 is the mean of first and second sample respectively,

μ1 and μ2 is the mean of first and second population respectively,

σX¯X¯ is the standard deviation and the formula to calculate σX¯X¯ is given by,

σX¯X¯=N1s12+N2s22N1+N22N1+N2N1N2

Where, N1 and N2 is the number of first and second population respectively.

Calculation:

As the significant difference in the sample statistics is to be determined and a direction has been predicted, a one tailed test is applied.

Follow the steps for two-sample testing as,

Step 1. Making assumptions and meeting test requirements.

Model:

Independent random samples.

Level of measurement is interval ratio.

Population Variance is Equal.

Sampling distribution is Normal.

Step 2. Stating the null hypothesis.

The statement of the null hypothesis is that there is no significant difference in the samples of the population. Thus, the null and the alternative hypotheses are,

H0:μ1=μ2

H1:μ1>μ2

Step 3. Selecting the sampling distribution and establishing the critical region.

Since, the sample size is small, t distribution can be used.

Thus, the sampling distribution is t distribution.

The level of significance is,

α=0.05

Degrees of freedom is,

N1+N22=43+372=78

Area of critical region is,

t(critical)=1.671

Step 4. Computing test statistics.

The population standard deviations are unknown.

The formula to calculate σX¯X¯ is given by,

σX¯X¯=N1s12+N2s22N1+N22N1+N2N1N2

Substitute 0

Expert Solution
To determine

(b)

To find:

If the senior citizens who live in retirement communities are more socially active over the phone than those who live in age integrated communities.

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