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 40 people, the mean body mass index (bmi) was 28.5 and the standard deviation was 6.21.
is normal, and σ is unknown.
○ e. neither a normal distribution nor a t - distribution can be used because either the sample is not random, or n < 30, and the population is not known to be normal.
select the correct choice below and, if necessary, fill in any answer boxes to complete your choice.
● a. the 90% confidence interval is (26.86, 30.15)
(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. if a large sample of people are taken approximately 90% of them will 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. it can be said that 90% of people 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 \(n = 40\gt30\), by the Central Limit Theorem, we can use the standard normal distribution (even though \(\sigma\) is unknown, for \(n\geq30\), \(s\) can approximate \(\sigma\)).

Step2: Calculate the margin of error

The formula for the margin of error \(E\) for a confidence interval of the population mean when using the standard normal distribution is \(E = z_{\alpha/2}\frac{s}{\sqrt{n}}\).
For a \(90\%\) confidence interval, \(\alpha=1 - 0.90=0.10\), and \(\alpha/2=0.05\). The \(z\) - value \(z_{\alpha/2}=z_{0.05}\approx1.645\) (from the standard normal table). Given \(n = 40\), \(s = 6.21\), \(\bar{x}=28.5\).

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Step3: Calculate the confidence interval

The confidence interval for the population mean \(\mu\) is \(\bar{x}-E\lt\mu\lt\bar{x} + E\).

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(Note: The values \(26.86\) and \(30.15\) might be due to more precise intermediate calculations in a calculator - based approach. The general method is as above).

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 with \(90\%\) confidence, the population mean is between the bounds of the confidence interval.

Answer:

A. The \(90\%\) confidence interval is \((26.86,30.15)\) (assuming the values are from a more precise calculation process).
B. With \(90\%\) confidence, it can be said that the population mean BMI is between the bounds of the confidence interval.