(a)
To Explain: that the state appropriate hypothesis for testing the company’s claim.
(a)
![Check Mark](/static/check-mark.png)
Answer to Problem R9.5RE
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:
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.
(b)
To Explain: a Type-I and Type-II error in this setting and give the consequences of each.
(b)
![Check Mark](/static/check-mark.png)
Explanation of Solution
Given:
From result part (a)
Type I error: Reject the null hypothesis
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
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 Explain: a significance level of 0.01, 0.05 or 0.10 for this test, justify the choice.
(c)
![Check Mark](/static/check-mark.png)
Answer to Problem R9.5RE
Explanation of Solution
Given:
From result part (a)
Type I error: Reject the null hypothesis
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
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
(d)
To Explain: the power of the test to detect the fact that only 3% of adults who use this vaccine would develop flu using
(d)
![Check Mark](/static/check-mark.png)
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
Explanation of Solution
Given:
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
(e)
To Explain: two ways that could increase the power of the test from part (d).
(e)
![Check Mark](/static/check-mark.png)
Answer to Problem R9.5RE
Increase significance level
Increase
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
Chapter 9 Solutions
PRACTICE OF STATISTICS F/AP EXAM
Additional Math Textbook Solutions
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Introductory Statistics
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