# In the 1970s a study was conducted in Philadelphia in which 500 cases were randomly assigned to treatments for the common cold: 250 subjects received

In the 1970s a study was conducted in Philadelphia in which 500 cases were randomly assigned to treatments for the common cold: 250 subjects received the medication and 250 received a placebo. A total of 383 patients improved within 24 hours. Of those who received the medication 241 improved within 24 hours and of those who received the placebo 142 improved within 24 hours. A test of significance was conducted on the following hypotheses.
${H}_{o}$: The rates for the two treatments are equal.
${H}_{a}$: The treatment of medication has a higher improvement rate.
This test resulted in a p-value of 0.0761.
a.) Interpret what this p-value measures in the context of this study.
b.) Based on this p-value and study design, what conclusion should be drawn in the context of this study? Use a significance level of $\alpha =0.05$.
c.) Based on your conclusion in part (b), which type of error, Type I or Type II, could have been made? What is one potential consequence of this error?
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Step 1
a)
The p-value is 0.0761.
Thus, the probability of obtaining a test statistic value at least as extreme as the observed value, when the rates of two treatments are equal, is 0.0761.
Step 2
b)
Null hypothesis:
${H}_{0}$: The rates of two treatments are equal.
Alternative hypothesis:
${H}_{a}$: The treatment of medication has a higher improvement rate.
Decision rule:
If p-value is less than or equal to level of significance, then reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
As p-value of 0.0761 is greater than 0.05, fail to reject the null hypothesis.
Hence, there is not enough evidence to claim that the treatment of medication has a higher improvement rate, at 0.05 level of significance.
c)
Type I error:
The type I error is the rejection of the null hypothesis when it is actually true.
Type II error:
The type II error is the failure of rejection of the null hypothesis when alternative hypothesis is true.
In this case, there is not enough evidence to claim that the treatment of medication has a higher improvement rate. However, there might be a chance that actually the treatment of medication has a higher improvement rate.
Hence there is a chance of type II error.
When type II error occurs in this case, one would believe that medication does not have a higher improvement rate, and would see no need of applying medication, when in reality medication leads to improved rate.