Q.38 - The Practice of Statistics for AP

Question

Aspirin prevents blood from clotting and so helps prevent strokes. The Second European Stroke Prevention Study asked whether adding another anticlotting drug named dipyridamole would be more effective for patients who had already had a stroke. Here are the data on strokes during the two years of the study: 

(a) Summarize these data in a two-way table.

 (b) Make a graph to compare the rates of strokes for the four treatments. Describe what you see. 

(c) Explain in words what the null hypothesis $H_0$: p1 = p2 = p3 = p4 says about the incidence of strokes. 

(d) Find the expected counts if H0 is true, and display them in a two-way table similar to the table of observed counts 

Step-by-Step Solution

Answer

a.

b.

(c) The null hypothesis states that the proportion of successes for each treatment is the same, based on the participants' smoking behaviours. 

(d)  The expected counts are the total of the rows multiplied by the total of the columns, divided by the sample size 

Part (a)Step 1: Given information

Given in the question that,  Aspirin prevents blood from clotting and so helps prevent strokes. The Second European Stroke Prevention Study asked whether adding another anticlotting drug named dipyridamole would be more effective for patients who had already had a stroke. Here are the data on strokes during the two years of the study: 

We need to summarize these data in a two-way table .

Part(a) Step 2: Explanation

Aspirin reduces blood coagulation and hence helps to avoid strokes, according to the question. As a result, the question includes statistics on strokes during the two-year research. Thus, in a two-way table, the number of no strokes equals the number of patients minus the number of strokes, that is,

	Stroke	No stroke
Placebo	$250$	$1399$	$1649$
Aspirin	$206$	$1443$	$1649$
Dipyridamole	$211$	$1443$	$1654$
Both	$157$	$1493$	$1650$
	$824$	$5778$	$6602$

Part(b) Step 1: Given information

Given in the question that, a table which contains data on strokes during the two years of the study: 

We need to make a graph to compare the rates of strokes for the four treatments. 

Part (b) Step 2: Explanation

Aspirin reduces blood coagulation and hence helps to avoid strokes, according to the question. As a result, the question includes statistics on strokes during the two-year research. Thus, in a two-way table, the number of no strokes equals the number of patients minus the number of strokes, that is,

As a result, the proportion in each category is computed by dividing the number of strokes/no strokes by the row total given in the last column:

	Stroke	No Stroke
Placebo	$0.1516$	$0.8484$
Aspirin	$0.1249$	$0.8751$
Dypyridamole	$0.1276$	$0.8724$
Both	$0.0952$	$0.9048$

As a result, the following graph can be used to compare the success rates of the four treatments:

Thus, by looking at the histogram we note that the proportions of strokes/no strokes do not seems to vary much by group.

Part(c) Step 1: Given information

Given in the question that,  Aspirin prevents blood from clotting and so helps prevent strokes. The Second European Stroke Prevention Study asked whether adding another anticlotting drug named dipyridamole would be more effective for patients who had already had a stroke. Here are the data on strokes during the two years of the study: 

We need to explain in words what the null hypothesis $H_0: p1 = p2 = p3 = p4$ says about the incidence of strokes.

Part(c) Step 2: Explanation

Aspirin reduces blood coagulation and hence helps to avoid strokes, according to the question. As a result, the question includes statistics on strokes during the two-year research. Thus, in a two-way table, the number of no strokes equals the number of patients minus the number of strokes, that is,

As a result, the null hypothesis in the question is written as follows: 

$H_0:p_1=p_2=p_3=p_4$

And the null hypothesis states that the proportion of successes for each treatment is the same, based on the participants' smoking behaviours.

Part(d) Step 1: Given information

Given in the question that,  Aspirin prevents blood from clotting and so helps prevent strokes. The Second European Stroke Prevention Study asked whether adding another anticlotting drug named dipyridamole would be more effective for patients who had already had a stroke. Here are the data on strokes during the two years of the study: 

We need to find the expected counts if $H_0$ is true, and

display them in a two-way table similar to the table of observed counts

Part (d) Step 2: Explanation

Aspirin reduces blood coagulation and hence helps to avoid strokes, according to the question. As a result, the question includes statistics on strokes during the two-year research. Thus, in a two-way table, the number of no strokes equals the number of patients minus the number of strokes, that is,

As a result, the null hypothesis in the question is written as:

$H_0:p_1=p_2=p_3=p_4$

As a result, the expected counts are the total of the rows multiplied by the total of the columns, divided by the sample size $n=6602$, as follows: