Question

A study was conducted to estimate hospital costs for accident victims who wore seat belts. Twenty randomly selected cases have a distribution that appears to be bell-shaped with a mean of $9004 and a standard deviation of $5629.

a) Construct the 99% confidence interval for the mean of all such costs and write a sentence that interprets the interval.

b) Based on the confidence interval, if you are a manager for an insurance company that provides lower rates for drivers who wear seat belts, and you want a conservative estimate for a worst case scenario, what amount should you use as the possible hospital cost for an accident victim who wears seat belts?

Answer 1

The 99% confidence interval for the mean hospital cost for accident victims who wore seat belts is between $5762.74 and $12245.26. For a conservative worst-case scenario estimate, an insurance company manager should use the higher end of the interval, $12245.26, for planning purposes.

Step-by-step explanation:

To construct the 99% confidence interval for the mean hospital costs for accident victims who wore seat belts, we utilize the sample mean, standard deviation, and the number of cases along with a Z-score that corresponds to the 99% confidence level. Since the distribution is approximately bell-shaped, we can assume that the sample distribution of the mean will be normal. For a 99% confidence interval and a sample size of 20, the critical value (Z-score) is approximately 2.576. Utilizing the formula for the confidence interval:

CI = mean ± (Z * (SD / sqrt(n)))

We find:

CI = $9004 ± (2.576 * ($5629 / sqrt(20)))

After calculation:

CI = $9004 ± (2.576 * (1258.49))

CI = $9004 ± $3241.26

Which gives us:

CI = ($5762.74, $12245.26)

The interpretation of this interval is that we are 99% confident that the true mean hospital cost for accident victims who wore seat belts is between $5762.74 and $12245.26.

For part b), if an insurance company manager is looking for a conservative estimate for the worst-case scenario regarding possible hospital costs, he should use the higher end of the confidence interval, which is $12245.26. This would allow the company to prepare for the higher potential costs that could be encountered.

Answer 2

12700$

Step-by-step explanation:

Given that sample size = 20, mean = 9004 and s= 5629

a) For 99% confidence interval, we use t distribution as population std dev is not known.

t critical value = 2.861

Margin of error = 2.861 *s/sqrt n

=
`2.869((5629)/(√(19) ) \\=3694.64`

Confidence interval lower bound

=
`9004-3694.64 = 5309.36`

Upper bound =
`9004+3694.64 = 12698.64`

b) Conservative estimate is the upper bound i.e. approxy 12700 dollars.