Question

To test the belief that sons are taller than their fathers are, a student are 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 = 0.01 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, Xi	Height of Son, Yi
72.3	77.4
73.1	76.6
68.2	70.6
67.1	68.8
67.7	68.8
67.1	67.8
67.8	67.7
69.9	69.4
71.5	70.4
67.7	66.0
72.4	69.9
73.2	69.7
69.8	64.7

Let di =   Xi - Yi. The hypotheses for the test Is:

H0​: μd = 0

H1​:  μd < 0

a) Calculate the test statistic:

t0 =

(Round to two decimal places)

(b) What is the P-value

P-value =

(Round to three decimal places)

c) Do not reject or Reject Upper H 0 because the​ P-value is ▼ less than or greater than the level of significance. There ▼ is or is not sufficient evidence to conclude that sons ▼ are the same height or are shorter than or are taller than or are not the same height as their fathers at the 0.01 level of significance.

Answer 1

Using Excel we get, n = 13

mean of di = 0 and standard deviation of di = 2.8501

The null and alternative hypotheses are,

H0 : μd = 0

H1 : μd < 0

a) test statistic is,

=> t_{​​​​​​0} = 0.00

b) degrees of freedom = 13 - 1 = 12

p-value = 0.500

c) Since, p-value = 0.500 > 0.01,

Do not reject H0, because p-value is greater than the level of significance. There is not sufficient evidence to conclude that sons are taller than their fathers at the 0.01 level of significance.

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