Questions: Widows A recent study indicated that 18% of the 196 women over age 55 in the study were widows. Round up your answers to the next whole number for the following questions. How large a sample must you take to be 90% confident that the estimate is within 0.05 of the true proportion of women over age 55 who are widows? n=

Solution Steps

We are tasked with determining the required sample size $n$ to estimate the proportion of women over age 55 who are widows with a confidence level of $90\%$. The following values are provided:

• Estimated proportion $p = 0.18$
• Margin of error $E = 0.05$
• Z-score for $90\%$ confidence level $Z \approx 1.645$

The formula for calculating the required sample size $n$ for estimating a population proportion is given by:

$$n = \frac{Z^2 \cdot p \cdot (1 - p)}{E^2}$$

Substituting the known values into the formula:

$$n = \frac{(1.645)^2 \cdot 0.18 \cdot (1 - 0.18)}{(0.05)^2}$$

Calculating each component:

• $(1.645)^2 \approx 2.706025$
• $1 - 0.18 = 0.82$
• $(0.05)^2 = 0.0025$

Thus, we have:

$$n = \frac{2.706025 \cdot 0.18 \cdot 0.82}{0.0025}$$

Calculating the numerator:

$$2.706025 \cdot 0.18 \cdot 0.82 \approx 0.4001$$

Now, substituting back into the equation for $n$:

$$n = \frac{0.4001}{0.0025} \approx 160.04$$

Since the sample size must be a whole number, we round up $160.04$ to the next whole number:

$$n = 161$$

Final Answer