Question

A 95% confidence interval for the average salary of all CEOs in the electronics industry wasconstructed using the results of a random survey of 45 CEOs. The interval was ($97,911, $110,014).To make more useful inferences from the data, it is desired to reduce the width of the confidenceinterval.

Which of the following will result in a reduced interval width?3

A) Increase the sample size and increase the confidence level.

B) Decrease the sample size and increase the confidence level.

C) Decrease the sample size and decrease the confidence level.

D) Increase the sample size and decrease the confidence level

Answer 1

D) Increase the sample size and decrease the confidence level

Explanation:

To build the confidence interval, initially we have to find the critical value of Z.

90% confidence interval

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`

95% confidence interval

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

`\alpha = (1-0.95)/(2) = 0.025`

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
`1-0.025 = 0.975`, so
`z = 1.96`

The width 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.

So the correct answer is:

D) Increase the sample size and decrease the confidence level