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Chapter 8: Q. 25 (page 393)

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

H0:渭=33lb(mean has not increased)
Ha:渭>33lb(mean has increased),

where 渭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 2010mean of 33lb.

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

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given Information 

The null and alternative hypotheses for the hypothesis test in Problem24:
H0:渭=33lb(mean has not increased)
Ha:渭>33lb{"x":[[5,4],[5,34],[35,33],[59,58,52,46,46,50,55,58,59,59,59,61],[71,72,72,72,72],[71],[114,114,114,115,119,127,135,139,140,140,140,140,144,150],[169,200,200,169],[217,246,247,217,218,230,241,246,245,233,221,215],[256,285,286,256,257,269,280,285,284,272,260,254],[329,326,335,344],[356,352,365,381,383,376,363,354]],"y":[[9,115],[52,52],[9,115],[103,95,91,98,116,123,124,114,92,91,119,123],[52,51,51,51,51],[116],[168,52,52,105,115,117,112,105,85,51,52,110,115,116],[68,82,83,95],[9,9,9,51,51,50,56,72,108,117,115,101],[9,9,9,51,51,50,56,72,108,117,115,101],[15,110,119,112],[9,114,117,108,70,52,51,59]],"t":[[0,0],[0,0],[0,0],[0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0],[0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0],[0,0,0,0,0,0,0,0]],"version":"2.0.0"}(mean has increased),

02

Step 2. For a.

In Type I error:
This error is committed in rejecting the null hypothesis(H0), when it is true.
We can come to the conclusion that the mean cheese consumption has increased from 2010mean of 33lbbut in reality the mean has not increased.

03

Step 3. For b.

Type II error:
This error is committed in accepting a null hypothesis (H0), when it is false. The conclusion is that the mean cheese consumption has not increased from the 2010mean of 33lbbut in reality the mean has increased from the 2010 mean of 33lb.

04

Step 4. For c.

The two ways of making the correct decisions are:

(i) If H0is true, then do not reject H0.

(ii) If H0is false, then reject H0.

Correct decision:

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

05

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 2010mean of 33lb. This signifies that the null hypothesis is not rejected. Therefore, Type I error committed in rejecting a null hypothesis H0, when it is true.

Correct decision:

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

06

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 2010mean of 33lb. This signifies that the null hypothesis is rejected. Therefore, Type II error committed in rejecting a null hypothesis H0, when it is true.

Correct decision:

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

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