Question

Suppose we want a 90% confidence interval for the average amount spent on books by freshmen in their first year at a major university. The interval is to have a margin of error of no more than $2. Based on last year's book sales, we estimate that the standard deviation of the amount spent will be close to $30. The number of observations required is closest to:

(a) 25.
(b) 30.
(c) 608.
(d) 609.
(e) 865.

Answer 1

Answer: option A

Step-by-step explanation: We can construct a (1 - α) % confidence interval for mean by using the formulae below.

u = x + critical value × (standard deviation /√n)

Where x = sample mean and n is the sample size.

The critical value that we are going to use (either z or t critical values) depends on the sample size.

If n > 29 we make use of a z test and if n < 29 we make use of a t test.

Also when using a z test we will (sometimes) make use of the population standard deviation.

And when using a t test, we make use of the sample standard deviation.

"Based on last year's book sales, we estimate that the standard deviation of the amount spent will be close to $30"

Judging by the sentence above, we can see that last year books sales is a fraction of several years of book sales, hence the value of standard deviation given is a sample standard deviation.

Since we are making use of a sample standard deviation, we will be using a t test for our critical value and hence sample size must be less than 29.

These points validates option A