 # As the director of the local Boys Club, you have claimed for years that membership in your club reduces juvenile delinquency. Now a cynical member of your funding agency has demanded proof of your claim. Random samples of members and non- members are gathered and interviewed with respect to their involvement in delinquent activities. Each respondent is asked to enumerate the number of delinquent acts he has engaged in over the past year. The average numbers of admitted acts of delinquency are reported below. What can you tell the funding agency? Sample 1 (Members) Sample 2 (Nonmembers) X ¯ 1 = 10.3 X ¯ 2 = 12.3 s 1 = 2.7 s 2 = 4.2 N 1 = 40 N 2 = 55 ### Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836 ### Essentials Of Statistics

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

#### Solutions

Chapter
Section
Chapter 8, Problem 8.8P
Textbook Problem
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## As the director of the local Boys Club, you have claimed for years that membership in your club reduces juvenile delinquency. Now a cynical member of your funding agency has demanded proof of your claim. Random samples of members and non- members are gathered and interviewed with respect to their involvement in delinquent activities. Each respondent is asked to enumerate the number of delinquent acts he has engaged in over the past year. The average numbers of admitted acts of delinquency are reported below. What can you tell the funding agency? Sample 1(Members) Sample 2(Nonmembers) X ¯ 1 = 10.3 X ¯ 2 = 12.3 s 1 = 2.7 s 2 = 4.2 N 1 = 40 N 2 = 55

Expert Solution
To determine

To find:

If the membership in the local boys club reduces juvenile delinquency.

### Explanation of Solution

Given:

The sample statistics is given in the table below,

 Sample 1(Members) Sample 2(Nonmembers) X¯1=10.3 X¯2=12.3 s1=2.7 s2=4.2 N1=40 N2=55

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=40+552=93

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 2

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