Questions: 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. (61.9,64.9) B. (60.8,65.4) C. (59.7,66.5) D. (58.1,67.3)

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. (61.9,64.9) B. (60.8,65.4) C. (59.7,66.5) D. (58.1,67.3)
Transcript text: 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 $\sigma$ is 2.4 and that the population of height measurements is normally distributed. Construct the $95 \%$ confidence interval for the population mean. A. $(61.9,64.9)$ B. $(60.8,65.4)$ C. $(59.7,66.5)$ D. $(58.1,67.3)$
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Solution

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Solution Steps

Step 1: Given Information

We have a sample of \( n = 10 \) randomly selected women with a mean height of \( \bar{x} = 63.4 \) inches. The population standard deviation is given as \( \sigma = 2.4 \) inches. We need to construct a \( 95\% \) confidence interval for the population mean height.

Step 2: Determine the Z-Score

For a \( 95\% \) confidence level, the significance level \( \alpha \) is \( 0.05 \). The critical value \( z \) corresponding to \( 95\% \) confidence is approximately \( 2.0 \).

Step 3: Calculate the Standard Error

The standard error (SE) of the mean is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{2.4}{\sqrt{10}} \approx 0.758 \]

Step 4: Calculate the Margin of Error

The margin of error (ME) is given by: \[ ME = z \cdot SE = 2.0 \cdot 0.758 \approx 1.516 \]

Step 5: Construct the Confidence Interval

The confidence interval is calculated as: \[ \bar{x} \pm ME = 63.4 \pm 1.516 \] This results in: \[ (63.4 - 1.516, 63.4 + 1.516) = (61.884, 64.916) \] Rounding to one decimal place gives us: \[ (61.9, 64.9) \]

Final Answer

The \( 95\% \) confidence interval for the population mean height is \\(\boxed{(61.9, 64.9)}\\).

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