Step 1: Conceptual Foundation:
A confidence interval delineates a range of probable values for a population parameter. The breadth of this interval is influenced by the chosen confidence level, the variability within the sample, and the sample's size. The formula for a confidence interval can be utilized to deduce the required sample size.
Step 2: Core Formula and Methodology:
The standard 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 defined as:\[ \text{ME} = Z_{\alpha/2} \times \frac{s}{\sqrt{n}} \]The sample mean (\(\bar{x}\)) and the margin of error can be extracted directly from the provided interval.
Step 3: Detailed Calculation:
The provided 95% confidence interval is [132, 160].
1. Determination of Sample Mean (\(\bar{x}\)): The sample mean is the central point of the confidence interval.\[ \bar{x} = \frac{\text{Upper Limit} + \text{Lower Limit}}{2} = \frac{160 + 132}{2} = \frac{292}{2} = 146 \]2. Calculation of Margin of Error (ME): The margin of error represents 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. Application of ME Formula to Determine n:Given values:- ME = 14- Sample standard deviation, s = 50- For a 95% confidence level, the critical Z-value, \(Z_{0.025}\), is 1.96.Substituting these values into the formula:\[ 14 = 1.96 \times \frac{50}{\sqrt{n}} \]Reordering the equation to isolate \(\sqrt{n}\):\[ \sqrt{n} = \frac{1.96 \times 50}{14} \]\[ \sqrt{n} = \frac{98}{14} = 7 \]Squaring both sides to ascertain n:\[ n = 7^2 = 49 \]Step 4: Conclusive Result:
The sample size employed in the study is 49.