Question

use the standard normal distribution or the t - distribution to construct a 90% confidence interval for the population mean. justify your decision. if neither distribution can be used, explain why. interpret the results.
in a random sample of 41 people, the mean body mass index (bmi) was 26.9 and the standard deviation was 6.01.
select the correct choice below and, if necessary, fill in any answer boxes to complete your choice.
a. the 90% confidence interval is (25.32, 28.48).
(round to two decimal places as needed.)
b. neither distribution can be used to construct the confidence interval.
interpret the results. choose the correct answer below.
a. it can be said that 90% of people have a bmi between the bounds of the confidence interval.
b. with 90% confidence, it can be said that the population mean bmi is between the bounds of the confidence interval.
c. if a large sample of people are taken approximately 90% of them will have a bmi between the bounds of the confidence interval.
d. neither distribution can be used to construct the confidence interval.

Explanation:

Step1: Determine the distribution

Since the sample size \(n = 41\) (\(n\geq30\)) and the population standard deviation \(\sigma\) is unknown, we use the \(t -\)distribution. The formula for the confidence interval for the population mean when using the \(t -\)distribution is \(\bar{x}\pm t_{\alpha/2}\frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(s\) is the sample standard deviation, \(n\) is the sample size, and \(t_{\alpha/2}\) is the critical value.

For a \(90\%\) confidence interval, \(\alpha=1 - 0.90=0.10\), and \(\alpha/2=0.05\). The degrees of freedom \(df=n - 1=41-1 = 40\). Using a \(t -\)table or technology, \(t_{0.05,40}\approx1.684\).

Step2: Calculate the margin of error

The sample mean \(\bar{x}=26.9\), the sample standard deviation \(s = 6.01\), and \(n = 41\). The margin of error \(E=t_{\alpha/2}\frac{s}{\sqrt{n}}=1.684\times\frac{6.01}{\sqrt{41}}\)

First, calculate \(\frac{6.01}{\sqrt{41}}\approx\frac{6.01}{6.403}\approx0.939\)

Then \(E=1.684\times0.939\approx1.58\)

Step3: Calculate the confidence interval

The confidence interval is \(\bar{x}-E<\mu<\bar{x} + E\)

\(26.9-1.58<\mu<26.9 + 1.58\)

\(25.32<\mu<28.48\)

Step4: Interpret the confidence interval

A confidence interval for the population mean gives a range of values within which we are confident the population mean lies. A \(90\%\) confidence interval means that if we were to construct many such intervals from different samples, approximately \(90\%\) of them would contain the true population mean.

Answer:

A. The \(90\%\) confidence interval is \((25.32,28.48)\)
B. With \(90\%\) confidence, it can be said that the population mean BMI is between the bounds of the confidence interval.