Q 9.26 - Elementary Statistics

Question

Refer to Exercise 9.18. Explain what each of the following would mean.

(a) Type I error.

(b) Type II error.

(c) Correct decision.

Now suppose that the results of carrying out the hypothesis test lead to the rejection of the null hypothesis. Classify that conclusion by error type or as a correct decision if in fact the mean age at diagnosis of all people with early-onset dementia.

(d) is 55 years old.

(e) is less than 55 years old.

Step-by-Step Solution

Answer

(a) Rejecting a Null Hypothesis, when it is true.

(b) Rejecting a Null Hypothesis, when $\left(H_0\right)$ is false.

(c) If the true null hypothesis is not rejected or a false null hypothesis is rejected.

(d) Correct Decision.

(e) Type II error.

Step 1. Given Information.

The null hypothesis is,

$H_0:\mu =55$ years.

The alternative hypothesis is,

$H_0:\mu <55$ years.

Part (a). Type I error.

According to the definition of the type I error it is to reject a null hypothesis when it is true. A type I error would occur in fact $H_0:\mu =55 years$ true, that is the mean age at diagnosis of all people with early-onset dementia is $55 years$old, but the results of the sampling lead to conclude that the mean age at diagnosis of all people with early-onset dementia is less than $55 years$ old.

Part (b). Type II error.

According to the definition of the type II error, it is to not reject a null hypothesis when it $\left(H_0\right)$ is false. A type II error would occur in fact $\mu =55 years$ is not to be rejected, but the results of the sampling fall to lead to conclude that the mean age at diagnosis of all people with early-onset dementia is $55$ years old.

Part (c). Correct Decision.

A correct decision would occur if the true null hypothesis is not rejected or a false null hypothesis is rejected. Here, in the fact the mean RDA of adult females is  and the results of the sampling do not lead to rejection, so is a correct decision; or the mean RDA of adult males is  and the results of the sampling lead to that conclusion.