A certain town is clustered around the town's largest employer, a paper mill. A researcher notices a large number of children with asthma and suspects the proportion of children with asthma in this town may be larger than the nationwide proportion of 7%.
(a) (2 pts) Write the null and alternative hypotheses.
(b) (3 pts) Suppose the researcher committed a Type I error. What did the researcher conclude and what was the reality regarding children with asthma?
(c) (3 pts) Suppose the researcher committed a Type II error instead. What did the researcher conclude and what was the reality regarding children with asthma?

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joaobezerra joaobezerra · Answer 1 · 2021-11-03T09:03:04+01:00

Using statistical concepts, it is found that:

a)

The null hypothesis is: [tex]H_0: p \leq 0.07[/tex]

The alternative hypothesis is: [tex]H_0: p > 0.07[/tex]

b)

The type I error means that the researcher concluded that the proportion in this town was above 7%, while in reality it wasn't.

c)

The type II error means that the researcher concluded that the proportion in this town was not above 7%, while in reality it was.

Item a:

At the null hypothesis, we test if the proportion is the nationwide proportion of 7% of less, thus:

[tex]H_0: p \leq 0.07[/tex]

At the alternative hypothesis, we test if the proportion is larger than the nationwide proportion, that is:

[tex]H_1: p > 0.07[/tex]

Item b:

A Type I error happens when a true null hypothesis is rejected.

Thus, the type I error means that the researcher concluded that the proportion in this town was above 7%, while in reality it wasn't.

Item c:

A Type II error happens when a false null hypothesis is not-rejected.

Thus, the type II error means that the researcher concluded that the proportion in this town was not above 7%, while in reality it was.

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