Q 9.24 - Elementary Statistics

Question

Refer to Exercise 9.16. 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 lactation period of grey seals.

(d) equals 23 days.

(e) differs from 23 days.

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) Type II Error.

Step 1. Given Information.

The Null Hypothesis is,

$H_0:\mu =23$.

The Alternative Hypothesis is,

$H_0:\mu \ne 23$.

Part (a). Type I error.

This is the error committed to rejecting a null hypothesis $\left(H_0\right)$ when it is true.

The conclusion is that the mean location period of grey seals differs from $23$ days but in reality, the mean location period of grey seals does not differ from $23$ days.

Part (b). Type II error.

This is the error committed to accepting a null hypothesis $\left(H_0\right)$ when it is false.

The conclusion is that the mean location period of grey seals does not differ from $23$ days but in reality, the mean location period of grey seals differs from $23$ days.

Part (c). Correct Decision.

The following are the two ways of making the correct decision:

1. If $H_0$ is true, then do not reject $H_0$.
2. If $H_0$ is false, then reject $H_0$.

If the mean location period of grey seals does not differs from $23$ days, $\mu =23 days$, then the null hypothesis is not rejected.

If the mean location period of grey seals differs from $23$ days, $\mu \ne 23 days$, then the null hypothesis is rejected.

Part (d). Equals 23 days.

In this situation, a Type I error is committed.

The null hypothesis is not rejected because the mean location period of grey seals equals to $23$days since the mean location period of grey seals equals to $23$days.

Part (e). Differs 23 days.

In this situation, a Type II error is committed.

The null hypothesis is rejected because the mean location period of grey seals is not equals to $23$days since the mean location period of grey seals differs from $23$ days.