Question

A researcher wants to compare the heights of males between generations to see if they differ. To do this, he samples random pairs of males who are at least 18 years old and their fathers. He then splits them into a sample of fathers and a sample of sons. Suppose that data were collected for a random sample of 11 pairs, where each difference is calculated by subtracting the height of the son from the height of the father. Assume that the heights are normally distributed. The test statistic is t≈1.971, α=0.05, the corresponding rejection regions are t<−2.228 and t>2.228, the null hypothesis is H0:μd=0, and the alternative hypothesis is Ha:μd≠0.

Select all that apply:
a. Reject the null hypothesis.
b. Fail to reject the null hypothesis.
c. The conclusion of the hypothesis test is that there is sufficient evidence to suggest that the heights of males between generations are different.
d. The conclusion of the hypothesis test is that there is insufficient evidence to suggest that the heights of males between generations are different.

Answer 1

The correct option is

Option A and Option C

Explanation:

From the question we are told that

The sample size of paired men is
`n_p   =  11`

The test statistics is t≈1.971

The significance level is α=0.05

The rejection region is t<−2.228 and t>2.228

The null hypothesis is
`H_o :\mu_ d=0`

The alternative hypothesis is
`H_a :\mu_ d \\e 0`

Generally the degree of freedom is mathematically represented as

`df  =  n_1 +  n_2 - 2`

Here
`n_1` is the sample size of father which is
`n_1 =  11`

`n_2` is the sample size of males who are at least 18 years old which is
`n_2 =  11`

So

`df  =  11 +  11 - 2`

=>
`df  =  20`

Generally the critical values of α=0.05 from the t- distribution table at a degree of freedom of
`df  =  20` for a two -tailed test is

`t_(0.05 , 20 ) =  \pm 2.08596345`

From the value obtained we see that the critical value is within the region of rejection hence

The decision rule is

Reject the null hypothesis

The conclusion is

The conclusion of the hypothesis test is that there is sufficient evidence to suggest that the heights of males between generations are different.