Question

Circle the correct answer : An article in the New York Times† reported that heart attack risk could be reduced by taking aspirin. This conclusion was based on a designed experiment involving both a control group of individuals that took a placebo having the appearance of aspirin but known to be inert and a treatment group that took aspirin according to a specified regimen. Subjects were randomly assigned to the groups to protect against any biases and so that probability-based methods could be used to analyze the data. Of the 11,034 individuals in the control group, 189 subsequently experienced heart attacks, whereas only 104 of the 11,037 in the: 1) aspirin / control group had a heart attack. The incidence rate of heart attacks in the treatment group was only about: 2) one -fifth / half / four-fifths / twice that in the control group.  One possible explanation for this result is chance variation—that aspirin really: 3) does / doesn't have the desired effect and the observed difference is just typical variation in the same way that tossing two identical coins would usually produce different numbers of heads. However, in this case, inferential methods suggest that chance variation by itself cannot adequately explain the magnitude of the observed difference. Please explain the calculations.

Answer 1

Solution:

The correct answers are as follows:

An article in the New York Times† reported that heart attack risk could be reduced by taking aspirin. This conclusion was based on a designed experiment involving both a control group of individuals that took a placebo having the appearance of aspirin but known to be inert and a treatment group that took aspirin according to a specified regimen. Subjects were randomly assigned to the groups to protect against any biases and so that probability-based methods could be used to analyze the data. Of the 11,034 individuals in the control group, 189 subsequently experienced heart attacks, whereas only 104 of the 11,037 in the: 1) aspirin group had a heart attack. The incidence rate of heart attacks in the treatment group was only about: 2) half that in the control group.  One possible explanation for this result is chance variation—that aspirin really: 3) doesn't have the desired effect and the observed difference is just typical variation in the same way that tossing two identical coins would usually produce different numbers of heads. However, in this case, inferential methods suggest that chance variation by itself cannot adequately explain the magnitude of the observed difference.

To explain that whether the there is actual difference between the two groups or the difference is by chance, we shall conduct a two sample proportion test based on the given data. We shall use significance level of 0.01 in the test.

The null and alternative hypotheses are as follows:

  i.e. The proportion of the people who had heart attacks in both of the groups is equal.

i.e. The proportion of the people who had heart attack in treatment group is less than the proportion of people who had heart attack in control group.

The test statistic is given as follows:

  

Where, p_{​​​​​​1} and p_{​​​​​​2} are sample proportions. n_{​​​​​​1} and n_{​​​​​​2} are sample sizes.

and Q = 1 - P

Sample proportion of heart attack in treatment group is,

Sample proportion of heart attack in control group is,

n_{​​​​​​1} = 11037 and n_{​​​​​​2} = 11034

The value of the test statistic is -5.0014.

Since, our test is left tailed test, therefore we shall obtain left tailed p-value for the test statistic, which is given as follows:

p-value = P(Z < value of the test statistic)

p-value = P(Z < -5.0014)

p-value = 0.0000

Significance level = 0.01

Since, p-value is less than the significance level of 0.01, therefore we shall reject the null hypothesis (H_{​​​​​​0}) at 0.01 significance level.

Conclusion: At significance level of 0.01, there is enough evidence to conclude that observed difference is actual and it is not by chance.