Mathematics, 25.02.2020 22:26 mhwalters

Random samples of 13 women and 11 men yielded the following scores on a test: Women: 70, 78, 62, 96, 75, 68, 41, 74, 80, 47, 73, 94, 65 Men: 72, 60, 52, 87, 66, 74, 95, 50, 81, 70, 72

Use a 0.05 significance level to test the claim that test scores for women have a larger standard deviation than test scores for men.

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Answers

Answer from: kayleetweedy1

$F=\frac{s^2_1}{s^2_2}=\frac{15.588^2}{13.754^2}=1.284$

$p_v =P(F_{12,10}1.284)=0.351$

And we can use the following excel code to find the p value:"=1-F.DIST(1.284,12,10,TRUE)"

Since the $p_v \alpha$ we have enough evidence to FAIL to reject the null hypothesis. And we can say that we don't have enough evidence to conclude that the standard deviation for women is larger than the standard deviation for men at 5% of significance.

Step-by-step explanation:

Data given and notation

Women : 70, 78, 62, 96, 75, 68, 41, 74, 80, 47, 73, 94, 65

Men: 72, 60, 52, 87, 66, 74, 95, 50, 81, 70, 72

Let 1 the index for women and 2 for men

We can calculate the sample deviation with the following formula:

$s= \frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}$

n_1 = 13 represent the sampe size for women

n_2 =11 represent the sample size for men

s_1 =15.588 represent the sample deviation for women

s_2 =13.754 represent the sample deviation for men

$\alpha=0.05$ represent the significance level provided

Confidence =0.95 or 95%

F test is a statistical test that uses a F Statistic to compare two population variances, with the sample deviations s1 and s2. The F statistic is always positive number since the variance it's always higher than 0. The statistic is given by:

$F=\frac{s^2_1}{s^2_2}$

Solution to the problem

System of hypothesis

We want to test if test scores for women have a larger standard deviation than test scores for men, so the system of hypothesis are:

H0: $\sigma^2_1 \leq \sigma^2_2$

H1: $\sigma^2_1 \sigma^2_2$

Calculate the statistic

Now we can calculate the statistic like this:

$F=\frac{s^2_1}{s^2_2}=\frac{15.588^2}{13.754^2}=1.284$

Now we can calculate the p value but first we need to calculate the degrees of freedom for the statistic. For the numerator we have n_1 -1 =13-1=12 and for the denominator we have n_2 -1 =11-1=10 and the F statistic have 12 degrees of freedom for the numerator and 10 for the denominator. And the P value is given by:

P value

Since we have a right tailed test the p value is given by:

$p_v =P(F_{12,10}1.284)=0.351$

And we can use the following excel code to find the p value:"=1-F.DIST(1.284,12,10,TRUE)"

Conclusion

Answer from: Quest

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