Q 9.25 - Elementary Statistics

Question

Refer to Exercise 9.17. 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 iron intake of all adult females under the age of 51 years.

(d) equals the RDA of 18 mg per day.

(e) is less than the RDA of 18 mg per day.

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) Type I error.

(e) Correct Decision.

Step 1. Given Information.

The Null Hypothesis is,

$H_0:\mu =18mg$.

The Alternative Hypothesis is,

$H_0:\mu <18mg$.

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 $H_0:\mu =18mg$ true, that is the RDA of iron for adult females of age $51$is $18$mg but the result of the sampling lead to conclude that the mean RDA of females of age $51$ is less than $18$ mg. 

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 $\mu =18mg$is not to be rejected, but the results of the sampling fall to lead to conclude that the mean RDA of females of age $51$ is less than $18$mg.$18$

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 $\mu =18mg$ and the results of the sampling do not lead to rejection, so is a correct decision; or the mean RDA of adult males is $\mu <18mg$ and the results of the sampling lead to that conclusion.

Part (d). Equals the RDA of 18 mg per day.

Here, the mean iron intake of adult females of $51$years old is $18 mg$ per day, and the results of a hypothesis test lead to rejection of the null hypothesis.

We are rejecting the true null hypothesis of $\mu =18mg$, where it is also obtained as a sampling result. So, we are committing a Type I error.

Part (e). Is less than the RDA of 18mg per day.

Here, the mean iron intake of adult females of $51$years old is less than$18mg$ per day, and the results of a hypothesis test lead to rejection of the null hypothesis.

As a sampling result, we obtain the mean RDA of adult females of $51$years old is less than $18mg$ per day, and we are rejecting the null hypothesis that the mean of adult females of $51$years old is less than $18 mg$per day. Therefore, our decision is a correct decision.