Question

Researchers were studying the effects of aerobic exercise on diastolic blood pressure in patients newly diagnosed with heart disease. The goal of this study is to identify changes in blood pressure in each patient after carrying out a new weekly aerobic exercise routine (given to each patient) for 6 months. Researchers measured the diastolic blood pressure of each patient at the beginning of the study (baseline) and 6 months after. The results of the study are listed in Table 2.

Your task for this assignment is to analyze the data in Table 2 and conduct hypothesis testing.

To complete this exercise, you must follow the steps of hypothesis testing (see Hypothesis Testing (PDF)). Make sure that in your Word document you demonstrate you followed each of the steps! Feel free to refer to your assignment for the Online Week 3 module if you need guiding questions to help you complete this part of the assignment.

Table 2.

Diastolic Blood Pressure (mm Hg) of 20 study participants at baseline and 6 months after starting a new weekly aerobic exercise program

ID number	Diastolic Blood Pressure (mm Hg) Baseline	Diastolic Blood Pressure (mm Hg) 6 months
001	88.0	83.0
002	80.0	77.0
003	84.0	72.5
004	79.0	75.0
005	74.5	70.0
006	57.0	63.0
007	98.0	90.0
008	70.5	67.5
009	63.0	74.0
010	77.5	67.5
011	84.0	88.5
012	79.0	71.0
013	83.0	77.5
014	71.5	71.0
015	91.0	85.5
016	88.0	78.5
017	90.5	87.5
018	105.0	88.5
019	96.5	90.5
020	76.0	75.0

Answer 1

As the data has been collected for same patient twice, so the data is not independent. The sample data here is for before and after treatment on same sample elements. Thus its a paired sample data.

Hence, we will conduct a paired sample t-test.

Now, as we want to check if the new exercise is effective in reducing the systolic blood pressure, so assuming the first set of data (before treatment) to be from population 1 and the another one (collected after 6 months) to be from population 2, we can conduct a one tailed hypothesis test as shown-

Define .

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Then, Null Hypothesis - H₀: 0

And, Alternate Hypothesis - H₁: > 0.

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Degrees of freedom of test = n -1

= 20-1 = 19.

Assuming significance level of = 0.05, the critical value of test statistic is -

So, we would reject the null hypothesis if the calculated value of test statistic = t > 1.729.

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The test statistic is given by -

Where, and are the sample mean and standard deviation of differences of the two observations.

The calculation is shown below in the table -

S.No.	X1	X2	Xd = X1 - X2	(Xd - Xd_bar)^2
1	88	83	5	0.7225
2	80	77	3	1.3225
3	84	72.5	11.5	54.0225
4	79	75	4	0.0225
5	74.5	70	4.5	0.1225
6	57	63	-6	103.0225
7	98	90	8	14.8225
8	70.5	67.5	3	1.3225
9	63	74	-11	229.5225
10	77.5	67.5	10	34.2225
11	84	88.5	-4.5	74.8225
12	79	71	8	14.8225
13	83	77.5	5.5	1.8225
14	71.5	71	0.5	13.3225
15	91	85.5	5.5	1.8225
16	88	78.5	9.5	28.6225
17	90.5	87.5	3	1.3225
18	105	88.5	16.5	152.5225
19	96.5	90.5	6	3.4225
20	76	75	1	9.9225
Sum			83	741.55

Note that -

And-

Thus the calculated test statistic is -

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As the calculated value of test statistic = t > 1.729, so it lies in the region of rejection.

Hence, we reject the null hypothesis and conclude that there is enough evidence in the data to support the claim that aerobic exercise is effective in reducing the systolic blood pressure for patients newly diagonised with heart disease.