Question

Cheese Consumption. The null and alternative hypotheses for the hypothesis test in Problem $24$are, respectively,

$H_0:\mu =33lb$(mean has not increased)
$H_a:\mu >33lb$(mean has increased),

where $\mu$is last year's mean cheese consumption for all Americans. 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 rejection of the null hypothesis. Classify that decision by error type or as a correct decision if in fact last year's mean cheese consumption

d. has not increased from the $2010$mean of $33lb$.

e. has increased from the $2010$mean of $33lb$.

Short answer

The explanation of what each of the following would mean has been shared below.

Step-by-step solution

Step 1. Given Information 

The null and alternative hypotheses for the hypothesis test in Problem$24$:
$H_0:\mu =33lb$(mean has not increased)
$H_a:\mu >33lb$(mean has increased),

Step 2. For a.

In Type I error:
This error is committed in rejecting the null hypothesis$\left(H_0\right)$, when it is true.
We can come to the conclusion that the mean cheese consumption has increased from $2010$mean of $33lb$but in reality the mean has not increased.

Step 3. For b.

Type II error:
This error is committed in accepting a null hypothesis $\left(H_0\right)$, when it is false. The conclusion is that the mean cheese consumption has not increased from the $2010$mean of $33lb$but in reality the mean has increased from the $2010$ mean of $33lb$.

Step 4. For c.

The two ways of making the correct decisions are:

(i) If $H_0$is true, then do not reject $H_0$.

(ii) If $H_0$is false, then reject $H_0$.

Correct decision:

If the mean cheese consumption has not increased from the $2010$mean of $33lb$, then the null hypothesis is not rejected. But if the mean cheese consumption has increased from the $2010$mean of $33lb$, then the null hypothesis is rejected.

Step 5. For d.

Explanation:

As per this situation, Type I error is committed.

Reason being:

From the given information, the mean cheese consumption has not increased from the $2010$mean of $33lb$. This signifies that the null hypothesis is not rejected. Therefore, Type I error committed in rejecting a null hypothesis $\left(H_0\right)$, when it is true.

Correct decision:

Here, the null hypothesis is not rejected because the mean cheese consumption has not increased from the $2010$mean of $33lb$.

Step 6. For e.

Explanation:

As per this situation, Type II error is committed.

Reason being:

From the given information, the mean cheese consumption has increased from the $2010$mean of $33lb$. This signifies that the null hypothesis is rejected. Therefore, Type II error committed in rejecting a null hypothesis $\left(H_0\right)$, when it is true.

Correct decision:

Here, the null hypothesis is rejected because the mean cheese consumption has increased from the $2010$mean of $33lb$.