Question

A mortgage firm estimates the true mean current debt of local homeowners, using the current outstanding balance of a random sample of 35 homeowners. They find a 95% confidence interval for the true mean current debt to be ($56,000, $120,000). Which of the following would correctly produce a confidence interval with a smaller margin of error than this 95% confidence interval?

a. Using the balances of older homeowners because they will have paid off more of their mortgage
b. Using the balances of 50 homeowners rather than 35, because using more data gives a smaller margin of error
c. Using the balances of only fifteen homeowners rather than 35, because there is likely to be less variation in fifteen balances than 35
d. Using 99% confidence, because then its more likely that the true mean is contained in the confidence interval

Answer 1

b. Using the balances of 50 homeowners rather than 35, because using more data gives a smaller margin of error

Explanation:

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

95% confidence interval

`\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`

99% confidence interval

`\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`

Now, find we find the width of our confidence interval M as such

`M = z*(\sigma)/(√(n))`

So, as n increases, the width of the confidence interval decreases.

From the examples above, we can also conclude that as the confidence level increases, so does the value of z, which means that the width of the interval increases.

Which of the following would correctly produce a confidence interval with a smaller margin of error than this 95% confidence interval?

For a smaller margin of error, we need to have a bigger sample size.

So the correct answer is:

b. Using the balances of 50 homeowners rather than 35, because using more data gives a smaller margin of error