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Do athletes in different sports vary in terms of their readiness for college? Reported here are college entrance exam scores of fictitious random samples of college basketball and football players and male and female athletes. Is there a significant difference between groups? Write a sentence or two explaining the results. a. Sample 1 (Basketball Players) Sample 2 (Basketball Players) X ¯ 1 = 460 X ¯ 2 = 442 s 1 = 92 s 2 = 57 N 1 = 102 N 2 = 117 b. Sample 1 (Males) Sample 2 (Females) X ¯ 1 = 452 X ¯ 2 = 480 s 1 = 88 s 2 = 75 N 1 = 107 N 2 = 105

BuyFind

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 8.3P
Textbook Problem
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Do athletes in different sports vary in terms of their readiness for college? Reported here are college entrance exam scores of fictitious random samples of college basketball and football players and male and female athletes. Is there a significant difference between groups? Write a sentence or two explaining the results.

a.

Sample 1
(Basketball Players)
Sample 2
(Basketball Players)
X ¯ 1 = 460 X ¯ 2 = 442
s 1 = 92 s 2 = 57
N 1 = 102 N 2 = 117

b.

Sample 1 (Males) Sample 2 (Females)
X ¯ 1 = 452 X ¯ 2 = 480
s 1 = 88 s 2 = 75
N 1 = 107 N 2 = 105

Expert Solution
To determine

(a)

To find:

The significant difference in the sample statistics of the two samples.

Explanation of Solution

Given:

The sample statistics is given in the table below,

Sample 1
(Basketball Players)
Sample 2
(Football Players)
X¯1=460 X¯2=442
s1=92 s2=57
N1=102 N2=117

The five step model for hypothesis testing:

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 is given by,

Z(obtained)=(X¯1X¯2)(μ1μ2)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¯=s21N11+s22N21

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

Calculation:

From the given information, the sample size of the first sample is 102, the sample size of the second sample is 117, the sample mean of the first sample is 460, the sample mean of the second sample is 442, the sample standard deviation of the first sample is 92, and the sample standard deviation of the second sample is 57.

As the significant difference in the sample statistics is to be determined, a two 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.

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 sample s of the population. Thus, the null and the alternative hypotheses are,

H0:μ1=μ2

H1:μ1μ2

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

Since, the sample size is large, Z distribution can be used.

Thus, the sampling distribution is Z distribution.

The level of significance is,

α=0.05

Area of critical region is,

Z(critical)=±1.96

Step 4. Computing test statistics.

The population standard deviations are unknown.

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

σX¯X¯=s21N11+s22N21

Substitute 92 for s1, 57 for s2, 102 for N1, and 117 for N2 in the above mentioned formula,

Expert Solution
To determine

(b)

To find:

The significant difference in the sample statistics of the two samples.

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