Question

in a random sample of 60 computers, the mean repair cost was $150 and assume that the population standard deviation is $36. construct a 90% confidence interval for the population mean.

a. ($138,$162)
b. ($537,$654)
c. ($141,$159)
d. ($142,$158)

Explanation:

Step1: Find the z - score

For a 90% confidence interval, the significance level \(\alpha=1 - 0.90=0.10\), and \(\alpha/2 = 0.05\). The z - score \(z_{\alpha/2}\) corresponding to a right - tail area of \(0.05\) is \(z_{0.05}\approx1.645\) (from the standard normal distribution table).

Step2: Calculate the margin of error \(E\)

The formula for the margin of error when the population standard deviation \(\sigma\) is known is \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\).
Given \(n = 60\), \(\sigma=36\), and \(z_{\alpha/2}=1.645\).

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Step3: Calculate the confidence interval

The confidence interval for the population mean \(\mu\) is \(\bar{x}-E<\mu <\bar{x} + E\), where \(\bar{x}=150\).
\(\bar{x}-E=150 - 7.65=142.35\approx142\)
\(\bar{x}+E=150 + 7.65=157.65\approx158\)

Answer:

D. (\$142,\$158)