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 $n = 96$.