Question

A study is conducted to compare the lengths of time required by men and women to assemble a certain product. Past experience indicates that the distribution of times for both men and women is approximately normal but the variance of the times for women is less than that for men. A random sample of times for 11 men and 14 women produced the following data:

Men:

n1= 11
s1= 6.1

Women:

n2= 14
s2= 5.3

Test the hypothesis that the variance for men is greater than for women. Use both p-value method and critical value approach.

Answer 1

Answer:

1.33 < 2.67 ; Fail to reject H0 at 0.05

Step-by-step explanation:

Given the data :

Men:

n1= 11

s1= 6.1

Women:

n2= 14

s2= 5.3

The hypothesis :

H0 : σ1² = σ2²

H1 : σ1² > σ2²

To calculate the test statistic ; we use th Ftest statistics ;

F statistic = Larger sample variance / Smaller sample variance

Fstatistic = s1² / s2² = 6.1² / 5.3² = 37.21/28.09 = 1.325

The F critical value at :

df numerator = n - 1 = 11 - 1 = 10

df denominator = n - 1 = 14 - 1 = 13

Using the F distribution table :

F critical = 2.671

Since

F statistic < F critical ; Fail to reject H0 at 0.05

Answer 2

We fail to reject the null hypothesis at significance level of H0 : s1² = s2²

For the men, we have:

• n1= 11
• s1= 6.1

For the women, we have:

• n2= 14
• s2= 5.3

The null and the alternate hypotheses are:

• Null hypothesis H0 : s1² = s2²
• Alternate hypothesis H1 : s1² > s2²

The numerator and the denominator degrees of freedom are calculated as:

$\mathbf{df = n -1}$

So, we have:

$\mathbf{df_1 = 11 -1}$

$\mathbf{df_1 = 10}$ ----- numerator

$\mathbf{df_2 = 14 -1}$

$\mathbf{df_2 = 13}$ ----- denominator

The test statistic of the f test is:

$\mathbf{t = \frac{s_1^2}{s_2^2}}$

So, we have:

$\mathbf{t = \frac{6.1^2}{5.3^2}}$

$\mathbf{t = \frac{37.21}{28.09}}$

$\mathbf{t = 1.325}$

The critical values at $\mathbf{t = 1.325}$ and the degrees of freedom is:

$\mathbf{F= 2.671}$

By comparison, 1.325 is less than 2.671.

Hence, we fail to reject the null hypothesis at H0 : s1² = s2²

Read more about hypothesis at:

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