Question

A researcher randomly selects 6 fathers who have adult sons and records the​ fathers' and​ sons' heights to obtain the data shown in the table below. Test the claim that sons are taller than their fathers at the alpha equals 0.10α=0.10 level of significance. The normal probability plot and boxplot indicate that the differences are approximately normally distributed with no outliers so the use of a paired​ t-test is reasonable

67.5	72.9	68.4	71.2	67.7	66.8
71.1	71.3	67.6	74.7	68.0	68.9

Father in the top row (in above graph)

son in the bottom row (in above graph)

Find the test static for T???

Find the P value ???

What is the correct conclusion?? Reject? Do not reject??

Answer 1

The data is:

Pair	Father	Son	Difference
1	67.5	71.1	-3.6
2	72.9	71.3	1.6
3	68.4	67.6	0.8
4	71.2	74.7	-3.5
5	67.7	68	-0.3
6	66.8	68.9	-2.1

Paired Sample t-test

For the score differences we have, mean is Dˉ=-1.1833, the sample standard deviation is sD=2.214, and the sample size is n=6. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μD =0 Ha: μD <0 This corresponds to a Left-tailed test, for which a t-test for two paired samples be used. (2a) Critical Value Based on the information provided, the significance level is α=0.1, and the degree of freedom is n-1=6-1=5. Therefore the critical value for this Left-tailed test is tc​=-1.4759. This can be found by either using excel or the t distribution table. (2b) Rejection Region The rejection region for this Left-tailed test is t<-1.4759 (3)Test Statistics The t-statistic is computed as follows: (4) The p-value The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true. In this case, the p-value is 0.1237 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that t=-1.3092 > tc​=-1.4759, it is then concluded that the null hypothesis is Not rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0.1237, and since p=0.1237>0.1, it is concluded that the null hypothesis is Not rejected. (6) Conclusion It is concluded that the null hypothesis Ho is Not rejected. Therefore, there is Not enough evidence to claim that the population mean μ1​ is less than μ2, at the 0.1 significance level.

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