Chapter PVI, Problem 17RE

To determine

To explain is it reasonable to assume that the two lists are still equivalent for purposes of the hearing test when there is background noise and base your decision on a confidence interval for the mean difference in the number of words people might misunderstand.

Answer to Problem 17RE

We are $95\%$ confident that the mean difference in words misunderstand is between $-3.76\text{ and 3}\text{.10}$ and it is reasonable to assume that the two lists are still equivalent for purposes of the hearing test when there is background noise.

Explanation of Solution

It is given in the question that fitting someone for a hearing aid requires assessing the patient's hearing ability. First we will check the conditions for inference as:

Random condition: It is satisfied as the sample that has been taken for the study is randomly selected.

Independent condition: It is satisfied as the subjects with normal hearing ability were independent of each other.

Normal condition: It is satisfied as the data shows no outlier.

Now, all the conditions for inference are satisfied. Thus, to find the confidence interval for mean we will use the calculator $\text{TI}-89$ . In the STAT Intsmenu, choose $4:2-$ SampTInt. You must specify if you are using data stored in two lists or if you will enter the means, standard deviations, and sizes of both samples. You must also indicate whether to pool the variances (when in doubt, say no) and specify the desired level of confidence. So, we have the list of the data as:

Subject	List A	List B
1	24	26
2	32	24
3	20	22
4	14	18
5	32	24
6	22	30
7	20	22
8	26	28
9	26	30
10	38	16
11	30	18
12	16	34
13	36	32
14	32	34
15	38	32
16	14	18
17	26	20
18	14	20
19	38	40
20	20	26
21	14	14
22	18	14
23	22	30
24	34	42

And $\alpha =0.05$ so by using the calculator $\text{TI}-89$ , the confidence interval can be calculated as:

  $(-3.76,3.10)$ .

Thus, we are $95\%$ confident that the mean difference in words misunderstand is between $-3.76\text{ and 3}\text{.10}$ . Since the confidence interval contains zero in it then we can say that the two tapes could be equivalent. Therefore, it is reasonable to assume that the two lists are still equivalent for purposes of the hearing test when there is background noise.