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
33	64
31	64
36	54
23	35
34	36
29	67
27	49
25	60
36	45
31	49
29	56
22	67
30	51
37	31
37	38

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:H0: Select an answer The median age of actresses is more than the median age of actors. The median of the differences is NOT zero. The median of the differences is zero. The median age of actresses is less than the median age of actors.

H1:H1: Select an answer The median of the differences is zero. The median age of actresses is more than the median age of actors. The median of the differences is NOT zero. The median age of actresses is less 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?  (Round to three decimal places.)

At a 0.025 significance, what is the conclusion about the null? Select an answer Reject the null hypothesis. Fail to reject the null hypothesis. Fail to support the null hypothesis. Support the null hypothesis.

What is the conclusion about the claim? Select an answer Support the claim that there is no difference in median age. There is insufficient evidence to support the claim that there is no difference in median age. Fail to reject the claim that there is no difference in median age 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−x1d=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=0H0: μd=0

H1:μd≠0H1:μd≠0

Actress Age	Actor Age
33	64
31	64
36	54
23	35
34	36
29	67
27	49
25	60
36	45
31	49
29	56
22	67
30	51
37	31
37	38

What is the critical value for this test? t=±t=±  (Round to three decimal places.)

What is the test statistic for this sample? t=t=  (Round to three decimal places.)

What is the p-value?  (Round to three decimal places.)

Conclusion about the null: Select an answer Reject the null hypothesis. Fail to support the null hypothesis. Support the null hypothesis. Fail to reject the null hypothesis.

Conclusion about the claim: Select an answer Support the claim that there is no mean difference in the ages. There is insufficient evidence to support the claim that there is no mean difference in the ages. Fail to reject the claim that there is no mean difference in the ages. 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

sign test

sample 1	sample 2	difference=sample1-sample 2
64	33	31
64	31	33
54	36	18
35	23	12
36	34	2
67	29	38
49	27	22
60	25	35
45	36	9
49	31	18
56	29	27
67	22	45
51	30	21
31	37	-6
38	37	1

Ho: The median of the differences is zero

H1:The median of the differences is NOT zero

Positive Signs: 14

Negative Signs:1

Total Signs:15

success is min(+ve and -ve sign) = 1

here, X~bin(15,0.5)

p-value = 2*P(X<1) = 2*[P(x=0)]

P ( X =    0   ) = C( 15   ,   0   )*   0.50   ^   0   *   0.50   ^ 15   =   0.0000

p-value = 2*(0.0000) = 0.0000

since, p-value <α=0.025,

reject the null hypothesis

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

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matched pair t test

Ho :   µd=   0
Ha :   µd ╪   0

Sample #1	Sample #2	difference , Di =sample1-sample2
64	33	31
64	31	33
54	36	18
35	23	12
36	34	2
67	29	38
49	27	22
60	25	35
45	36	9
49	31	18
56	29	27
67	22	45
51	30	21
31	37	-6
38	37	1

	sample 1	sample 2	Di
sum =	766	460	306
mean=	51.0667	30.6667	20.4000

α=0.025
df=n-1=14

t-critical value , t* = ± 2.51 [from t table]

mean of difference ,    D̅ =   20.4000
      
std dev of difference , Sd =        14.807334
      
std error , SE =    Sd / √n =    3.8232
      
t-statistic =    (D̅ - µd)/SE =    5.336

p-value =        0.000

  p-value <α , Reject null hypothesis  

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