Question

a simple random sample of size 48 was taken, with a sample mean of 35.5, and a sample standard deviation of 5.6. compute the confidence intervals for the population mean at the following confidence levels. (round your answers to three decimal places.)
a. 90% confidence interval:

b. 95% confidence interval:

c. 99% confidence interval:

d. how do the margin of error and the width of the confidence interval change when the confidence level increases?
as the confidence level increases, the margin of error becomes larger, resulting in a wider confidence interval.
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Explanation:

Step1: Calculate the standard error

The formula for the standard error \(SE=\frac{s}{\sqrt{n}}\), where \(s = 5.6\) and \(n=48\).

$$SE=\frac{5.6}{\sqrt{48}}\approx\frac{5.6}{6.928}\approx0.808$$

Step2: Find the z - values for each confidence level

• For a \(90\%\) confidence level, the z - value \(z_{0.90}\): The significance level \(\alpha=1 - 0.90=0.10\), and \(\alpha/2 = 0.05\). From the standard normal table, \(z_{0.05}\approx1.645\)
• For a \(95\%\) confidence level, the z - value \(z_{0.95}\): The significance level \(\alpha=1 - 0.95 = 0.05\), and \(\alpha/2=0.025\). From the standard normal table, \(z_{0.025}\approx1.96\)
• For a \(99\%\) confidence level, the z - value \(z_{0.99}\): The significance level \(\alpha=1 - 0.99=0.01\), and \(\alpha/2 = 0.005\). From the standard normal table, \(z_{0.005}\approx2.576\)

Step3: Calculate the confidence intervals

The formula for the confidence interval is \(\bar{x}\pm z\times SE\), where \(\bar{x} = 35.5\) and \(SE\approx0.808\)

• For \(90\%\) confidence interval:

$$35.5-1.645\times0.808=35.5 - 1.330=34.170$$

$$35.5 + 1.645\times0.808=35.5+1.330 = 36.830$$

• For \(95\%\) confidence interval:

$$35.5-1.96\times0.808=35.5-1.584 = 33.916$$

$$35.5 + 1.96\times0.808=35.5 + 1.584=37.084$$

• For \(99\%\) confidence interval:

$$35.5-2.576\times0.808=35.5-2.081=33.419$$

$$35.5 + 2.576\times0.808=35.5+2.081 = 37.581$$

Answer:

a. \(90\%\) Confidence Interval: \(34.170\) to \(36.830\)
b. \(95\%\) Confidence Interval: \(33.916\) to \(37.084\)
c. \(99\%\) Confidence Interval: \(33.419\) to \(37.581\)
d. As the confidence level increases, the margin of error becomes larger, resulting in a wider confidence interval.