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.
which distribution should be used to construct the confidence interval? choose the correct answer below.
○ a. use a t - distribution because the sample is random, n ≥ 30, and σ is unknown.
○ b. use a normal distribution because the sample is random, the population is normal, and σ is known.
○ c. use a normal distribution because the sample is random, n ≥ 30, and σ is known.
○ d. use a t - distribution because the sample is random, the population 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 (, )
(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

Explanation:

Step1: Determine the distribution

Since the sample is random, \(n = 40\geq30\), and the population standard deviation \(\sigma\) is unknown (we are given the sample standard deviation \(s = 6.21\)), we use the t - distribution.

Step2: Find the critical value

For a \(90\%\) confidence interval and \(n-1=40 - 1=39\) degrees of freedom. Using a t - table or calculator, the critical value \(t_{\alpha/2}\) with \(\alpha=1 - 0.90 = 0.10\) and \(\alpha/2=0.05\) is approximately \(t_{0.05,39}\approx1.685\)

Step3: Calculate the margin of error

The formula for the margin of error \(E=t_{\alpha/2}\frac{s}{\sqrt{n}}\). Substituting \(t_{\alpha/2}=1.685\), \(s = 6.21\), and \(n = 40\)

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Step4: Calculate the confidence interval

The formula for the confidence interval is \(\bar{x}-E<\mu<\bar{x} + E\). Given \(\bar{x}=28.5\)

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Answer:

A. The \(90\%\) confidence interval is \((26.85,30.15)\)