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 alphaequals0.10 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
72.4 77.5
70.6 74.1
73.1 75.6
69.9 71.7
69.4 70.5
69.4 69.9
68.1 68.2
68.9 68.2
70.5 69.3
69.4 67.7
69.5 67
67.2 63.7
70.4 65.5
Which conditions must be met by the sample for this​ test? Select all that apply.
A. The sample size is no more than​ 5% of the population size.
B. The differences are normally distributed or the sample size is large.
C. The sample size must be large.
D. The sampling method results in a dependent sample.
E. The sampling method results in an independent sample.
Write the hypotheses for the test. Upper
H 0 ​:
H 1 ​:
Calculate the test statistic. t 0=? ​
(Round to two decimal places as​ needed.)
Calculate the​ P-value. ​P-value=?
​(Round to three decimal places as​ needed.) Should the null hypothesis be​ rejected?
▼ 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 0.10 level of significance. Click to select your answer(s).

Answer 1

Answer:

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

C. The  sample size mus be large

E. The sampling method results in an independent sample

2) The null hypothesis H₀:  $\bar x_1$ =  $\bar x_2$

The alternative hypothesis Hₐ: $\bar x_1$ <  $\bar x_2$

Test statistic, t = -0.00693

p- value = 0.498

Do not reject Upper H₀ because, the P-value is greater than the level of significance. There is sufficient evidence to conclude that sons are the same height as their fathers  at 0.10 level of significance

Step-by-step explanation:

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

C. The  sample size mus be large

E. The sampling method results in an independent sample

2) The null hypothesis H₀:  $\bar x_1$ =  $\bar x_2$

The alternative hypothesis Hₐ: $\bar x_1$ <  $\bar x_2$

The test statistic for t test is;

$t=\dfrac{(\bar{x}_1-\bar{x}_2)}{\sqrt{\dfrac{s_{1}^{2} }{n_{1}}-\dfrac{s _{2}^{2}}{n_{2}}}}$

The mean

Height of Father, h₁,  Height of Son h₂

72.4,      77.5

70.6,      74.1

73.1,       75.6

69.9,      71.7

69.4,      70.5

69.4,      69.9

68.1,       68.2

68.9,      68.2

70.5,       69.3

69.4,       67.7

69.5,       67

67.2,       63.7

70.4,       65.5

$\bar x_1$  = 69.6      

s₁ = 1.58

$\bar x_2$ = 69.9

s₂ = 3.97

n₁ = 13

n₂ = 13

$t=\dfrac{(69.908-69.915)}{\sqrt{\dfrac{3.97^{2}}{13}-\dfrac{1.58^{2} }{13}}}$

(We reversed the values in the square root of the denominator therefore, the sign reversal)

t = -0.00693

p- value = 0.498 by graphing calculator function

P-value > α Therefore, we do not reject the null hypothesis

Do not reject Upper H₀ because, the P-value is greater than the level of significance. There is sufficient evidence to conclude that sons are the same height as their fathers  at 0.10 lvel of significance