Question

The following data set shows the ages of the Best Actress and Best Actor award at a given awards show for various years:

Actress Age	Actor Age
24	44
36	36
24	40
37	54
22	50
37	53
24	44
27	43
32	65
36	44
32	57

Using a Sign Test, test the claim that there is no median difference between the ages of Best Actress and Best Actor award winners.

Find the null and alternative hypothesis.

H0:A)The median of the differences is NOT zero.

B)The median of the differences is zero.

C)The median age of actresses is more than the median age of actors.

D)The median age of actresses is less than the median age of actors.

H1:A)The median age of actresses is less than the median age of actors.

   B)The median of the differences is zero.

   C)The median of the differences is NOT zero.

D)The median age of actresses is more than the median age of actors.

If we consider + to represent when the female was older than the male, then how many of each sign is there?

Positive Signs:

Negative Signs:

Total Signs:

What is the p-value?

At a 0.025 significance, what is the conclusion about the null? A)Reject the null hypothesis.

B)Support the null hypothesis.

C)Fail to reject the null hypothesis.

   D)Fail to support the null hypothesis.

What is the conclusion about the claim? A)There is insufficient evidence to support the claim that there is no difference in median age.

   B)Support the claim that there is no difference in median age.

C)Fail to reject the claim that there is no difference in median age

   D)Reject the claim that there is no difference in median age.

Let's now perform a mean-matched pairs test to test the claim that there is no mean difference between the age of males and females. For the context of this problem, d=x2−x1 where the first data set represents actress (female) ages and the second data set represents male (actor) ages. We'll continue to use a significance of 0.025. You believe the population of difference scores is normally distributed, but you do not know the standard deviation.

H0: μd=0

H1:μd≠0

Actress Age	Actor Age
24	44
36	36
24	40
37	54
22	50
37	53
24	44
27	43
32	65
36	44
32	57

What is the critical value for this test? t=±

What is the test statistic for this sample? t=

What is the p-value?

Conclusion about the null: A)Support the null hypothesis.

   B)Reject the null hypothesis.

   C)Fail to reject the null hypothesis.

   D)Fail to support the null hypothesis.

Conclusion about the claim: A)Support the claim that there is no mean difference in the ages.

B)There is insufficient evidence to support the claim that there is no mean difference in the ages.

   C)Reject the claim that there is no mean difference in the ages.

D)Fail to reject the claim that there is no mean difference in the ages.

How were these two tests similar?

How were these two tests different?

Answer 1

Ho: The median of the differences is zero

Hence, B) is the correct option.

Ha: The median of the differences is not zero.

Hence, C) is the correct option.

Actress Age	Actor Age	Sign
24	44	-
36	36	NA
24	40	-
37	54	-
22	50	-
37	53	-
24	44	-
27	43	-
32	65	-
36	44	-
32	57	-

Positive signs: 0

Negative signs: 10

Total signs: 10

p-value = 0.00157, Hence we reject the null hypothesis at significance level of 0.025. A) is the correct option. We conclude that the median of the differences is not zero.

We reject the claim that there is no difference in the median age. Hence, D) is the correct option.

Matched-pairs test:

H0: μd=0

H1:μd≠0

Actress Age	Actor Age	d (Difference)
24	44	-20
36	36	0
24	40	-16
37	54	-17
22	50	-28
37	53	-16
24	44	-20
27	43	-16
32	65	-33
36	44	-8
32	57	-25

Critical value for the test at df = 11 – 1 = 10 is +2.634 and -2.634

We will use the following formula for the t-statistic:

d-bar = -18.09

sd = 9.05

n = 11

Hence, t = -6.629

p-value = 0.000059 (<0.001)

Conclusion about the null: B) We reject the null hypothesis

Conclusion about the claim: C) reject the claim that there is no mean difference in the ages

The two tests were similar in the sense that both were matched paired tests and both tested the difference in the samples.

The two tests were different in the sense that in sign test, we can’t quantify that by how much amount there is a difference, we just use signs to show the difference but in the second test we can quantify. Also, the sign test tells us the difference in the medians but the latter the difference in the means.