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. Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers

Height of Father,71.9,71.8,68.5,73.2,70.9,69.8,69.3,72.3,71.9,70.2,69.8,69.8,69.1 Xi Height of Son, Yi 76.8, 75.2, 71.0, 75.0, 72.0, 70.3, 69.4, 71.7, 70.6, 68.4, 67.3, 66.3, 64.2

Estimate the average difference in heights between fathers and sons.

a. What type of procedure should you do?

b. Construct a 90% confidence interval for the average difference in heights between fathers and sons.

c. Interpret the interval, if appropriate.

d. What is the margin of error of the interval?

Answer 1

a. A paired sample t-test will be most effective here. This is because we have a dependent dataset of father and son.

b.

Subject.	Xi Height of Father	Yi Height of Son	Difference
1	71.9	76.8	-4.9
2	71.8	75.2	-3.4
3	68.5	71	-2.5
4	73.2	75	-1.8
5	70.9	72	-1.1
6	69.8	70.3	-0.5
7	69.3	69.4	-0.1
8	72.3	71.7	0.6
9	71.9	70.6	1.3
10	70.2	68.4	1.8
11	69.8	67.3	2.5
12	69.8	66.3	3.5
13	69.1	64.2	4.9

Paired Sample t test Confidence Interval Output

For the score differences we have, mean is Dˉ=0.0231, the sample standard deviation is sD=2.8036, and the sample size is n=13. and the required confidence level is 90%. Degree of freedom The number of degrees of freedom are df = 13 - 1 = 12, and the significance level is α=0.1. Critical Value Based on the provided information, the critical t-value for α=0.1 and df=12 degrees of freedom is tc​=1.7823. Therefore, based on the information provided, the 90% confidence for the population mean μ is calculated as:

c. Therefore, the 90% confidence interval for the population mean difference μD is -1.3628<μ<1.409, which indicates that we are 90% confident that the true population mean difference μD between father and son heights is contained by the interval (-1.3628,1.409)

d. Margin of Error

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