Question

If a\(95\% \)confidence interval for the population mean was reported to be 160 to 170 and\(σ = 25\), then the size of the sample used in this study is:

• A. 96
• B. 125
• C. 54
• D. 81

Hint:

When calculating the sample size for a confidence interval, the margin of error formula \( E = Z \cdot \frac{\sigma}{\sqrt{n}} \) is very useful. The key step is isolating \( n \) after finding the margin of error. Be sure to square your result for \( \sqrt{n} \) to get the sample size. Also, remember to round the sample size to the nearest whole number since you cannot have a fraction of a sample.

Answer 1

The Correct Option is A

The formula for the margin of error in a confidence interval is:

\(E = Z \cdot \frac{\sigma}{\sqrt{n}},\)

where \(E = \text{Width of the interval} \div 2\), \(Z = 1.96\), and \(\sigma = 25\).

The width of the confidence interval is:  

\(170 - 160 = 10 \quad \Rightarrow \quad E = \frac{10}{2} = 5.\)
Substitute into the formula:  

\(5 = 1.96 \cdot \frac{25}{\sqrt{n}}.\)

Solve for \( n \):  

\(\sqrt{n} = \frac{1.96 \cdot 25}{5} = 9.8 \quad \Rightarrow \quad n = 9.8^2 = 96.04.\)

Thus, the sample size is \(n = 96\).

Answer 2

The formula for the margin of error in a confidence interval is:

\[ E = Z \cdot \frac{\sigma}{\sqrt{n}}, \] where: - \( E \) is the margin of error (half the width of the confidence interval), - \( Z = 1.96 \) (the Z-score corresponding to a 95% confidence level), - \( \sigma = 25 \) (the population standard deviation), - \( n \) is the sample size.

Step 1: Calculate the margin of error \( E \):

The width of the confidence interval is given by: \[ 170 - 160 = 10, \] So, the margin of error is: \[ E = \frac{10}{2} = 5. \]

Step 2: Substitute into the margin of error formula:

Substitute \( E = 5 \), \( Z = 1.96 \), and \( \sigma = 25 \) into the formula: \[ 5 = 1.96 \cdot \frac{25}{\sqrt{n}}. \]

Step 3: Solve for \( n \):

First, isolate \( \frac{25}{\sqrt{n}} \) by dividing both sides by 1.96: \[ \frac{25}{\sqrt{n}} = \frac{5}{1.96} \approx 2.55. \] Now, solve for \( \sqrt{n} \): \[ \sqrt{n} = \frac{25}{2.55} \approx 9.8. \]

Step 4: Find \( n \):

Square both sides to get: \[ n = 9.8^2 = 96.04. \]

Conclusion: Since the sample size must be an integer, round \( n = 96.04 \) to the nearest whole number: \[ n = 96. \] Thus, the required sample size is \( n = 96 \).