Answer :
Answer:
[tex]F=\frac{s^2_1}{s^2_2}=\frac{15.588^2}{13.754^2}=1.284[/tex]
[tex]p_v =P(F_{12,10}>1.284)=0.351[/tex]
And we can use the following excel code to find the p value:"=1-F.DIST(1.284,12,10,TRUE)"
Since the [tex]p_v > \alpha[/tex] 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:
[tex] s= \frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}[/tex]
[tex]n_1 = 13 [/tex] represent the sampe size for women
[tex]n_2 =11[/tex] represent the sample size for men
[tex]s_1 =15.588 [/tex] represent the sample deviation for women
[tex]s_2 =13.754 [/tex] represent the sample deviation for men
[tex]\alpha=0.05[/tex] 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:
[tex]F=\frac{s^2_1}{s^2_2}[/tex]
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: [tex] \sigma^2_1 \leq \sigma^2_2[/tex]
H1: [tex] \sigma^2_1 >\sigma^2_2[/tex]
Calculate the statistic
Now we can calculate the statistic like this:
[tex]F=\frac{s^2_1}{s^2_2}=\frac{15.588^2}{13.754^2}=1.284[/tex]
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 [tex]n_1 -1 =13-1=12[/tex] and for the denominator we have [tex]n_2 -1 =11-1=10[/tex] 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:
[tex]p_v =P(F_{12,10}>1.284)=0.351[/tex]
And we can use the following excel code to find the p value:"=1-F.DIST(1.284,12,10,TRUE)"
Conclusion
Since the [tex]p_v > \alpha[/tex] 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.