Chapter 9, Problem R9.5RE

(a)

To determine

To Explain: that the state appropriate hypothesis for testing the company’s claim.

Answer to Problem R9.5RE

  $H_0:p=0.05$

  $H_0:<0.05$

Explanation of Solution

Given:

Claim is that the proportion is fewer than 5%

The null hypothesis statement is that the population value is equal to the value given in the claim:

  $H_0:p=5\%=0.05$

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis statement is that the population proportion is equal to the value given in the claim. If the null hypothesis is the claim, then the alternative hypothesis statement is the opposite of the null hypothesis.

  $H_0:<0.05$

  $p$ is the population proportion of adults who use the vaccine get the flu.

(b)

To determine

To Explain: a Type-I and Type-II error in this setting and give the consequences of each.

Explanation of Solution

Given:

From result part (a)

  $\begin{array}{c}{H}_0:p=0.05\\ H_1:p<0.05\end{array}$

Type I error: Reject the null hypothesis $H_0$ , once the null hypothesis $H_0$ is true.

There is enough convincing evidence that the proportion of adults who received vaccines and who get the flu is less than 0.05, when the proportion of adults who received vaccines and who get the flu is actually 0.05. A possible consequence is that the adults receive the vaccine, while the vaccine is not effective and therefore the adults could still get sick.

Type II error: Fail to reject the null hypothesis $H_0$ , once the null hypothesis $H_0$ is false and there is no enough convincing proof that the proportion of adults who received vaccines and who get the flu if less than 0.05, when the proportion of adults who received vaccines and who get the flu is actually less than 0.05.

A possible consequence is that the vaccine is not made available to people, whereas the vaccine is actually effective and thus we missing out on an effective vaccine.

(c)

To determine

To Explain: a significance level of 0.01, 0.05 or 0.10 for this test, justify the choice.

Answer to Problem R9.5RE

  $\alpha =0.01$

Explanation of Solution

Given:

From result part (a)

  $\begin{array}{c}{H}_0:p=0.05\\ H_1:p<0.05\end{array}$

Type I error: Reject the null hypothesis $H_0$ , once the null hypothesis $H_0$ is true.

There is enough convincing proof that the proportion of adults who received vaccines and who get the flu is less than 0.05, when the proportion of adults who received vaccines and who get the flu is actually 0.05. A possible consequence is that the adults receive the vaccine, while the vaccine is not effective and therefore the adults could still get sick.

Type II error: Fail to reject the null hypothesis $H_0$ , once the null hypothesis $H_0$ is false

There is no enough convincing proof that the proportion of adults who received vaccines and who get the flu if less than 0.05, when the proportion of adults who received vaccines and who get the flu is actually less than 0.05. A possible consequence is that the vaccine is not made available to people, whereas the vaccine is actually effective and thus we missing out on an effective vaccine.

It is observed that a type I error is worse, as people think they receive a vaccine but are still likely to get the flu. Since a type I error is worse, we want to minimize the probability of a type I error, we should choose the smallest significance level, which is therefore $\alpha =0.01$ .

(d)

To determine

To Explain: the power of the test to detect the fact that only 3% of adults who use this vaccine would develop flu using $\alpha =0.05$ is 0.9437, interpret this value.

Answer to Problem R9.5RE

If the proportion of adults who use the vaccine and get the flu is 0.03 then have a 94.37% probability that find convincing proof to help the alternative hypothesis $H_1:p<0.05$ .

Explanation of Solution

Given:

  $\begin{array}{l}{H}_0:p=5\%=0.05\\ H_1:p<0.05\\ p_A=3\%=0.03\\ n=1000\\ \alpha =0.05\\ \text{Power=0}\text{.9437=94}\text{.37\%}\end{array}$

The power is the probability of rejecting the null hypothesis once the alternative hypothesis is true. If the proportion of adults who use the vaccine and get the flu is 0.03 then have a 94.37% probability that find convincing proof to help the alternative hypothesis $H_1:p<0.05$ .

(e)

To determine

To Explain: two ways that could increase the power of the test from part (d).

Answer to Problem R9.5RE

Increase significance level

Increase sample size

Making the alternative proportion p more extreme

Explanation of Solution

The power is the probability of rejecting the null hypothesis once the alternative hypothesis is true.

Increase the power by:

Increasing the sample size the reason is that it is having more data about the population will allow making better estimations

Increasing the significance level the reason is that this increases the probability of making a Type I error and decreases the probability of making a Type II error.

Making the alternative proportion $p$ more extreme, therefore it is lesser although more extreme alternatives are easier to prove.