Question

in a sample of 10 randomly selected women, it was found that their mean height was 63.4 inches. from previous studies, it can be assumed that the population standard deviation σ is 2.4 and that the population of height measurements is normally distributed. construct the 95% confidence interval for the population mean.

a. (59.7,56.5)
b. (61.9,64.9)
c. (60.8,65.4)
d. (58.1,67.3)

Explanation:

Step1: Find the z - score

For a 95% confidence interval, the z - score \(z_{\alpha/2}\) is 1.96.

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

The formula for the margin of error when the population standard deviation \(\sigma\) is known is \(E = z_{\alpha/2}\times\frac{\sigma}{\sqrt{n}}\).
Given \(n = 10\), \(\sigma=2.4\), and \(z_{\alpha/2}=1.96\).

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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 \(\bar{x}=63.4\).

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Answer:

B. \((61.9,64.9)\)