You may have heard of the Beck Depression Inventory (BDI). This is a 21-question survey where higher total scores are indicative of more severe depressive symptoms. In particular, scores greater than 30 may suggest severe depression.

Suppose a psychologist wants to investigate whether retired adults are more likely to suffer from depression. A random sample of 18 retired adults was given the BDI and their total scores are presented in Table 4.8.

a. Find and interpret a \(95 \%\) confidence interval for the population mean BDI.

b. Do you think that retired adults have a mean BDI total score greater than 30 ?

c. How many adults would have to be sampled to detect a difference of at least 33 on the BDI?

Table 4.8

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BDI Scores for a Random Sample of 18 Retired Adults 28 29 22 29 30 30 31 32 39 21 29 32 33 28 28 28 25 39 38