Question

Forty-five CEO’s from the electronics industry were randomly sampled and a 99 % % confidence interval for the average salary of all electronics CEO’s was constructed. The interval was $101,866<μ<$115,016 $101,866<μ<$115,016 To make more useful inferences from the data, it is desired to reduce the width of the confidence interval. What will result in a reduced interval width? Select one: a. Any of these methods will result in a reduced interval width. b. Keep the sample size the same and decrease the confidence level. c. Keep the sample size the same and increase the confidence level. d. Decrease the sample size and keep the same confidence level.

Answer 1

b. Keep the sample size the same and decrease the confidence level.

Explanation:

We first have to find the critical value of z, which depends of the confidence level.

90% confidence level

We have that to find our
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1-0.9)/(2) = 0.05`

Now, we have to find z in the Ztable as such z has a pvalue of
`1-\alpha`.

So it is z with a pvalue of
`1-0.05 = 0.95`, so
`z = 1.645`

99% confidence level

We have that to find our
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1-0.99)/(2) = 0.005`

Now, we have to find z in the Ztable as such z has a pvalue of
`1-\alpha`.

So it is z with a pvalue of
`1-0.005 = 0.995`, so
`z = 2.575`

The width of the interval is:

`W = z*(\sigma)/(√(n))`

In which
`\sigma` is the standard deviation of the population and n is the size of the sample.

So, as z increses, so does the width. If z decreases, the width decreases. Lower confidence levels have lower values of z.

As n increases, the width decreses.

What will result in a reduced interval width?

b. Keep the sample size the same and decrease the confidence level.