Question

If a 95% confidence interval for a population mean was reported to be 132 to 160 and sample standard deviation s = 50, then the size of the sample in the study is:
(Given \(Z_{0.025}\) = 1.96)

• A. 90
• B. 95
• C. 50
• D. 49

Hint:

The margin of error is the key link between the confidence interval, standard deviation, and sample size. If you are given the interval, you can always calculate the ME and then use its formula to solve for any unknown component.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A confidence interval provides a range of plausible values for a population parameter. The width of this interval depends on the confidence level, the sample variability, and the sample size. We can use the formula for the confidence interval to work backward and solve for the sample size.
Step 2: Key Formula or Approach:
The formula for a confidence interval for the mean is: \[ \text{CI} = \bar{x} \pm \text{Margin of Error (ME)} \] Where the Margin of Error is given by: \[ \text{ME} = Z_{\alpha/2} \times \frac{s}{\sqrt{n}} \] We can find the sample mean (\(\bar{x}\)) and the margin of error from the given interval.
Step 3: Detailed Explanation:
The given 95% confidence interval is [132, 160].
1. Calculate the sample mean (\(\bar{x}\)): The sample mean is the midpoint of the confidence interval. \[ \bar{x} = \frac{\text{Upper Limit} + \text{Lower Limit}}{2} = \frac{160 + 132}{2} = \frac{292}{2} = 146 \] 2. Calculate the Margin of Error (ME): The margin of error is half the width of the interval. \[ \text{ME} = \frac{\text{Upper Limit} - \text{Lower Limit}}{2} = \frac{160 - 132}{2} = \frac{28}{2} = 14 \] 3. Use the ME formula to find n: We have: - ME = 14 - Sample standard deviation, s = 50 - For a 95% confidence level, the critical Z-value is \(Z_{0.025}\) = 1.96. Now, substitute these values into the formula: \[ 14 = 1.96 \times \frac{50}{\sqrt{n}} \] Rearrange the equation to solve for \(\sqrt{n}\): \[ \sqrt{n} = \frac{1.96 \times 50}{14} \] \[ \sqrt{n} = \frac{98}{14} = 7 \] Finally, square both sides to find n: \[ n = 7^2 = 49 \] Step 4: Final Answer:
The size of the sample in the study is 49.