Question

To test the belief that sons are taller than their​ fathers, a student randomly selects 13 fathers who have adult male children. She records the height of both the father and son in inches and obtains the following data. Are sons taller than their​ fathers? Use the alpha=0.025 level of significance.​ Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers.

Height of Father   Height of Son
71.4 76.3
72.8   76.3
71.9   74.5
72.4   74.2
70.3   71.6
71.7   72.3
73.1   73.2
73.1   72.4
69.2   68.0
66.7   64.9
67.8   65.3
68.4   64.9
69.2   64.3

Which conditions must be met by the sample for this​ test? Select all that apply.

A.

The differences are normally distributed or the sample size is large.

B.

The sample size is no more than​ 5% of the population size.

C.

The sampling method results in a dependent sample.

D.

The sample size must be large.

E.

The sampling method results in an independent sample.

Let

d Subscript idiequals=Upper X Subscript iXiminus−Upper Y Subscript iYi.

Write the hypotheses for the test.

Upper H 0H0​:

mu Subscript d Baseline equals 0μd=0

mu Subscript d Baseline greater than 0μd>0

mu Subscript d Baseline equals 0μd=0

mu Subscript d Baseline less than 0μd<0

mu Subscript d Baseline not equals 0μd≠0

Upper H 1H1​:

mu Subscript d Baseline greater than 0μd>0

mu Subscript d Baseline greater than 0μd>0

mu Subscript d Baseline less than 0μd<0

mu Subscript d Baseline not equals 0μd≠0

mu Subscript d Baseline equals 0μd=0

Calculate the test statistic.

t 0t0equals=negative 0.02−0.02

​(Round to two decimal places as​ needed.)

Calculate the​ P-value.

​P-valueequals=0.4920.492

​(Round to three decimal places as​ needed.)

Should the null hypothesis be​ rejected?

Do not reject

Reject

Do not reject

Upper H 0H0

because the​ P-value is

greater than

less than

greater than the level of significance. There is not sufficient evidence to conclude that sons are taller than are not the same height as are taller than are shorter than are the same height as their fathers at the0.025level of significance.

Answer 1

Correct options are A, B and C.

Let d = height of father - height of son

Following table shows the calculations:

Height of father	Height of son	d	(d-mean)^2
71.4	76.3	-4.9	23.863225
72.8	76.3	-3.5	12.145225
71.9	74.5	-2.6	6.682225
72.4	74.2	-1.8	3.186225
70.3	71.6	-1.3	1.651225
71.7	72.3	-0.6	0.342225
73.1	73.2	-0.1	0.007225
73.1	72.4	0.7	0.511225
69.2	68	1.2	1.476225
66.7	64.9	1.8	3.294225
67.8	65.3	2.5	6.325225
68.4	64.9	3.5	12.355225
69.2	64.3	4.9	24.157225
Total		-0.2	95.996925

Should the null hypothesis be​ rejected?

Do not reject H0 because the​ P-value is greater than the level of significance.

Conclusion: There is not sufficient evidence to conclude that sons are taller than their fathers at the 0.025 level of significance.