Question

Stress and heart attacks You read a newspaper article that describes a study of whether stress management can help reduce heart attacks. The $107$subjects all had reduced blood flow to the heart and so were at risk of a heart attack. They were assigned at random to three groups. The article goes on to say:

One group took a four-month stress management program, another underwent a four-month exercise program, and the third received usual heart care from their personal physicians. In the next three years, only $3$of the $33$people in the stress management group suffered "cardiac events," defined as a fatal or non-fatal heart attack or a surgical procedure such as a bypass or angioplasty. In the same period, $7$of the $34$people in the exercise group and $12$out of the $40$patients in usual care suffered such events.

a. Use the information in the news article to make a two-way table that describes the study results.

b. Compare the success rates of the three treatments in preventing cardiac events.

c. Do the data provide convincing evidence at the $\alpha =0.05$level that the true success rates for patients like these are not the same for the three treatments?

Short answer

(a) The two-way table is

	Cardiac Event	No Cardiac Event
Stress Management	$3$	$30$	$33$
Exercises	$7$	$27$	$34$
Usual Care	$12$	$28$	$40$
	$22$	$85$	$107$

(b) The success rate of the treatment is its rate of stress management, exercises, and usual care.

(c) No, the data is not proving convincing evidence for the true success rates in patient treatment.

Step-by-step solution

Part (a) Step 1: Given information

We need to find the two way table.

Part (a) Step 2: Explanation

We know that

• Each combination of care and the cardiac event is counted in the two-way table (or not).
• The total of each row is shown in the last column.
• The total of each column is shown in the last row.

	Cardiac event	No cardiac event
Stress Management	$3$	$30$	$33$
Exercise	$7$	$27$	$34$
Usual care	$12$	$28$	role="math" $40$
	$22$	$85$	$107$

Part (b) Step 1: Given information

We need to find the comparison of the success rates of the three treatments.

Part (b) Step 2: Explanation

The success rate is calculated by dividing the number of "no cardiac events" by the entire row count:

• Stress management is $\frac{33}{100}=0.9091$
• Exercise is $\frac{27}{34}=0.7941$
• Usual care is$\frac{28}{40}=0.7$

Part (c) Step 1: Given information

We need to find out whether the data is convincing evidence or not.

Part (c) Step 2: Explanation

We know that

The null hypothesis asserts that the variables are unrelated, whereas the alternative hypothesis asserts that they are.

$H_0$is the true success rate for patients having the same for three treatments.

$H_{\alpha }$ is the true success rate for patients not having the same for three treatments.

And expected frequencies are a product of row and column total divided by table total.

And The squared differences between the actual and predicted frequencies, divided by the expected frequency, make up the chi-square subtotals.

Therefore, the data is not proving convincing evidence for the true success rates in patient treatment.