Final answer:
The correct 94.4% confidence interval for the population mean, given a sample mean of 24, a sample size of 121, and a known population standard deviation of 5.3, is approximately 23.234 to 24.766, option (b). This is calculated using the formula for a confidence interval with a known population standard deviation and normal distribution.
Step-by-step explanation:
The question seeks to find a 94.4% confidence interval for the population mean given a random sample of 121 observations with known mean, median, and mode values. When the population standard deviation is known, we can use the normal distribution to compute the confidence interval. The confidence interval formula for a known population standard deviation is:
CI = μ ± (Z* · (σ/√n))
Where:
• μ is the sample mean
• Z* is the z-value that corresponds to the desired confidence level
• σ is the population standard deviation
• n is the sample size
In this case, the sample mean (μ) is 24, n is 121, and the population standard deviation (σ) is 5.3. The confidence level is 94.4%, which corresponds to a z-value approximately 1.86. Plugging these into the formula:
CI = 24 ± (1.86 · (5.3/√121))
Calculating the margin of error, EBM:
EBM = 1.86 · (5.3/√121) ≈ 1.86 · 0.481 ≈ 0.895
The confidence interval is:
CI = 24 ± 0.895 ≈ (23.105, 24.895)
Thus, the correct answer is (b) 23.234 to 24.766, which is rounded to two decimal places as the population standard deviation is given with one decimal place precision.