Question

Question 3 A study was performed to determine whether men and women differ in their repeatability in assembling components on printed circuit boards. Random samples of 25 men and 21 women were selected, and each subject assembled the units. The two sample standard deviations of assembly time were s Subscript men Baseline equals 0.98 minutes and s Subscript women Baseline equals 1.02 minutes. (a) Is there sufficient evidence to support the claim that men and women differ in repeatability for this assembly task at the 0.02 level of significance

Answer 1

To determine if men and women differ in repeatability for an assembly task, a two-sample t-test can be conducted using the given sample standard deviations. The test statistic is compared to the critical value from the t-distribution to make a conclusion.

Step-by-step explanation:

To test whether there is sufficient evidence to support the claim that men and women differ in repeatability for this assembly task, we can conduct a two-sample t-test. First, we calculate the pooled standard deviation, which is a weighted average of the sample standard deviations. The formula is:

pooled standard deviation = sqrt(((n1-1)*s1^2 + (n2-1)*s2^2) / (n1+n2-2))

Using the given information, the pooled standard deviation is:

pooled standard deviation = sqrt(((25-1)*(0.98)^2 + (21-1)*(1.02)^2) / (25+21-2))

Next, we calculate the test statistic, which measures the difference between the sample means:

test statistic = (sample mean 1 - sample mean 2) / (pooled standard deviation * sqrt(1/n1 + 1/n2))

Finally, we compare the test statistic to the critical value from the t-distribution with (n1 + n2 - 2) degrees of freedom at a significance level of 0.02. If the test statistic is greater than the critical value, we can reject the null hypothesis and conclude that there is sufficient evidence to support the claim that men and women differ in repeatability for this assembly task.

Answer 2

There is no sufficient evidence to support the claim that men and women differ in repeatability for this assembly task

Step-by-step explanation:

Given

Let subscript 1 represent men and 2 represent women, respectively.

`n_1 = 25`

`n_2 = 21`

`s_1 = 0.98`

`s_2 = 1.02`

`\alpha = 0.02`

Required

Determine if here is enough evidence

First, we need to state the hypotheses

`H_o: \sigma_1^2 = \sigma_2^2`

`H_1: \sigma_1^2 \\e \sigma_2^2`

Next, calculate the test statistic using:

`F = (s_1^2)/(s_2^2)`

`F = (0.98^2)/(1.02^2)`

`F = 0.923`

Calculate the rejection region;

But first, calculate the degrees of freedom

`df_1 =n_1 - 1`

`df_1 =25 - 1`

`df_1 =24`

`df_2 = n_2 -1`

`df_2 = 21 - 1`

`df_2 = 20`

Using the F Distribution: table

`c = (\alpha)/(2)`

`c = (0.02)/(2)`

`c = 0.01`

At 0.01 level (check row 20 and column 24), the critical value is:

`f_(0.01,24,20) = 2.86` --- the upper bound

At 0.01 level (check row 24 and column 20), the critical value is:

`f_(0.01,20,24) = 2.74`

Calculate the inverse F distribution.

`f_(0.99,20,24) = (1)/(f_(0.01,20,24)) = (1)/(2.74) =0.365` ---- the lower bound

The rejection region is then represented as:

`0.365 < Test\ Statistic < 2.86`

If the test statistic falls within this region, then the null hypothesis is rejected

`F = 0.923` --- Test Statistic

`0.365 < 0.923 < 2.86`

The above inequality is true; so, the null hypothesis is rejected.

This implies that, there is no sufficient evidence.