Step 1: Concept Definition:
A confidence interval indicates a range of likely values for a population parameter. The width of this range is influenced by the confidence level, sample variability, and sample size. The sample size can be determined by rearranging the confidence interval formula.
Step 2: Core Formula:
The formula for a confidence interval for the mean is:\[ \text{CI} = \bar{x} \pm \text{Margin of Error (ME)} \]The Margin of Error is calculated as:\[ \text{ME} = Z_{\alpha/2} \times \frac{s}{\sqrt{n}} \]From the provided interval, both the sample mean (\(\bar{x}\)) and the margin of error can be derived.
Step 3: Calculation Breakdown:
Given the 95% confidence interval: [132, 160].
1. Determine the sample mean (\(\bar{x}\)): This is the midpoint of the interval.\[ \bar{x} = \frac{160 + 132}{2} = 146 \]2. Calculate the Margin of Error (ME): This is half the interval's width.\[ \text{ME} = \frac{160 - 132}{2} = 14 \]3. Solve for n using the ME formula:Known values:- ME = 14- Sample standard deviation, s = 50- For a 95% confidence level, \(Z_{0.025}\) = 1.96.Substitute into the formula:\[ 14 = 1.96 \times \frac{50}{\sqrt{n}} \]Isolate \(\sqrt{n}\):\[ \sqrt{n} = \frac{1.96 \times 50}{14} = 7 \]Square both sides to find n:\[ n = 7^2 = 49 \]Step 4: Conclusion:
The sample size for this study is 49.