Thensive Quiz 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. 12 possible A. \( (61.9,64.9) \) B. \( (58.1,67.3) \) C. \( (60.8,65.4) \) D. \( (59.7,66.5) \)

To construct a \( 95 \% \) confidence interval for the population mean, we can use the formula for the confidence interval: \[ \text{CI} = \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right) \] Where: - \( \bar{x} = 63.4 \) (sample mean) - \( \sigma = 2.4 \) (population standard deviation) - \( n = 10 \) (sample size) - \( z \) is the z-value corresponding to the desired confidence level (approximately 1.96 for \( 95 \% \) confidence) Calculating the standard error (SE): \[ SE = \frac{2.4}{\sqrt{10}} \approx 0.76 \] Now, using the z-score to calculate the margin of error: \[ \text{Margin of Error} = 1.96 \times 0.76 \approx 1.49 \] Now, calculate the confidence interval: \[ CI = 63.4 \pm 1.49 = (61.91, 64.89) \] Rounding to one decimal place, the \( 95 \% \) confidence interval is approximately \( (61.9, 64.9) \). So the answer is **A. \( (61.9, 64.9) \)**. --- Confidence intervals give you a range of plausible values for the population mean based on your sample data. It's like having a safety net—if you were to repeat the process many times, you expect the true mean to fall within this range in \( 95 \% \) of the cases! Understanding how these intervals work in practice can be crucial, especially in fields like medicine or economics, where decisions depend on estimating values based on sample data. They're not just numbers; they represent the uncertainty and reliability of our estimates!