Chapter B, Problem 1E

a.

To determine

State the conclusion of the test.

Answer to Problem 1E

The conclusion of the test is Reject $H_0$.

Explanation of Solution

Here, denote ${\mu }_1$ as the mean recovery time of students with colds who are given large doses of vitamin C and ${\mu }_2$ as the mean recovery time of students with colds who are not given vitamin C.

The test hypotheses are given below:

Null hypothesis: $H_0:{\mu }_1={\mu }_2$

That is, the mean recovery time is the same for the students with colds who are given large doses of vitamin C and not given vitamin C.

Alternative hypothesis: $H_a:{\mu }_1>{\mu }_2$

That is, the mean recovery time of students with colds who are given large doses of vitamin C is faster than those who are not given vitamin C.

Rejection rule:

If the $\text{p-value}\le \text{significance level}$, reject the null hypothesis.

If the $\text{p-value}>\text{significance level}$, do not reject the null hypothesis.

Conclusion:

Here, the p-value of 0.003 is less than the 0.05 significance level.

From the rejection rule, the null hypothesis is rejected.

Hence, the conclusion is that there is evidence to conclude that the mean recovery time of students with colds who are given large doses of vitamin C is faster than those who are not given vitamin C at 5% of significance level.

b.

To determine

Delineate an inappropriate method of collecting the data that would bias the results so much that a conclusion based on the p-value is very unreliable.

Explanation of Solution

The answers will vary. One of the possible answers is given below:

• Give vitamin C for the students who have had cold symptoms for a long time.
• The latter may have shorter recovery times since the cold is almost over when they start treatment, while the students those who are not getting vitamin C might be at the early stages of their colds.

c.

To determine

Delineate an appropriate method of collecting the data that would permit to interpret the p-value to extend the results to a broader student population.

Explanation of Solution

The answers will vary. One of the possible answers is given below:

• Randomize the students into two groups (using lot system or flip a coin)
• Flip a coin to specify who gets vitamin C (heads) and who gets the placebo (tails).
• Neither the participants nor the person determining when they recovered from cold must know the group they belong to.

d.

To determine

Make a conclusion about vitamin C as a treatment for the students with common cold using the p-value from the assumed data in Part (c).

Explanation of Solution

If the data are collected in the same way explained in Part (c), then it is assumed that the p-value obtained is small.

The small p-value indicates that the null hypothesis $H_0:{\mu }_1={\mu }_2$ is in favor of the alternative $H_a:{\mu }_1<{\mu }_2$. Therefore, there is strong evidence that the mean recovery time of students with colds who are given large doses of vitamin C is faster than those who are not given vitamin C.