Question

Data are shown for a study of pulse rates of students sitting versus standing. At α = 0.05, test the claim that the standing pulse rate is higher than the sitting pulse rate for students.

Student	Sitting Pulse Rate (bpm), x	Standing Pulse Rate (bpm), y
1 2 3 4 5 6 7 8 9 10 11 12 13 14	74 74 58 80 78 62 74 62 68 64 60 56 52 80	78 76 60 96 90 64 74 70 66 74 80 58 52 88

Answer 1

The data is:

Student	Sitting Pulse Rate (bpm), x	Standing Pulse Rate (bpm), y	Difference
1	74	78	-4
2	74	76	-2
3	58	60	-2
4	80	96	-16
5	78	90	-12
6	62	64	-2
7	74	74	0
8	62	70	-8
9	68	66	2
10	64	74	-10
11	60	80	-20
12	56	58	-2
13	52	52	0
14	80	88	-8

Sample Size (n)	14
Sample Mean(Xˉ)	-6.000
Sample St. Deviation (s)	6.563

Paired Sample t-test

For the score differences we have, mean is Dˉ=-6, the sample standard deviation is sD=6.5633, and the sample size is n=14. (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=0.05, and the degree of freedom is n-1=14-1=13. Therefore the critical value for this Left-tailed test is tc​=-1.7709. 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<-1.7709 (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 0.0023 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that t=-3.4205 < tc​=-1.7709, it is then concluded that the null hypothesis is rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0.0023, and since p=0.0023≤0.05, it is concluded that the null hypothesis is rejected. (6) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ1​ is less than μ2, at the 0.05 significance level.

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