Statistics finding the confidence level of the interval

Petrolovujhm

Petrolovujhm

Answered question

2022-04-03

Statistics finding the confidence level of the interval estimate medical expenses
A confidence interval for the true mean of the annual medical expenses of a middle-class American family is given as (738, 777). If this interval is based on interviews with 110 families and a standard deviation of $ 120 is assumed. Suppose all annual medical expenses of middle-class American families follow an approximately normal distribution.
(a) What is the sample mean of annual medical expenses?
Sample mean =777+7382=757.5
(b) What is the confidence level of the interval estimate (as a decimal)
CI=sample mean+zα2(standard deviationsquare root n)
Isolate for z α2
777=757.5+zα2(120square root 110)
Rearrange
777square root110757.5120=zα2
0.089650657=zα2
What are the correct solutions and answers?

Answer & Explanation

Jadyn Gentry

Jadyn Gentry

Beginner2022-04-04Added 12 answers

Foremost, you are told that the standard deviation of the expenses is assumed to be σ=120. This is your first hint that the confidence interval was derived using a normal distribution, not a student's t distribution, because the latter uses the sample variance s that is calculated from the data.
Thus, you have correctly inferred that the interval has the form
[738,777]=[L,U]=[xzα2σn,;x+zα2σn].
It follows that
x=L+U2=738+7772
as you again correctly compute. It also follows that
zα2σn=UL2,
which is a quantity usually given the name "margin of error." Hence the critical value
zα2=nσ×UL21.70431,
and if you recall, this means Pr[Z>zα2]=Pr[Z>1.70431]=α2,
where Z is the standard normal distribution. So you would either use a calculator or a normal distribution table to find that α20.0441612, which means that the provided confidence interval corresponds to a significance level of about α=0.088, or 100(1α)%=91.17% confidence.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in College Statistics

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?