Questions: A company manufactures goods that are sold exclusively by mail order. The director of market research needed to test market a new product. She planned to send brochures to a random sample of households and use the proportion of orders obtained as an estimate of the true proportion, known as the product response rate. The results of the market research were to be utilized as a primary source for advance production planning, so the director wanted the figures she presented to be as accurate as possible. Specifically, she wanted to be 95% confident that the estimate of the product response rate would be accurate to within 2%. Complete parts (a) through (d). a. Without making any assumptions, determine the sample size required. n = (Round up to the nearest whole number.) b. Historically, product response rates for products sold by this company have ranged from 0.5% to 6.7%. If the director had been willing to assume that the sample product response rate for this product would also fall in that range, find the required sample size. n= (Round up to the nearest whole number.) c. Compare the results from parts (a) and (b). Using a likely range from part (b) for p the required sample size. d. Discuss the possible consequences if the assumption made in part (b) turns out to be incorrect.

Solution Steps

To determine the sample size required without making any assumptions about the product response rate, we use the formula for sample size estimation for a proportion:

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

Where:

• \( Z \) is the Z-score for a 95% confidence level, which is approximately \( 1.96 \).
• \( p \) is the estimated proportion, and in the absence of any assumptions, we use the worst-case scenario \( p = 0.5 \).
• \( E \) is the margin of error, which is \( 0.02 \).

Substituting the values:

\[ n = \frac{(1.96)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.02)^2} = \frac{(3.8416) \cdot 0.5 \cdot 0.5}{0.0004} = \frac{0.9604}{0.0004} = 2401 \]

Thus, the sample size required without assumptions is:

\[ \boxed{n = 2401} \]

Historically, the product response rates for this company have ranged from \( 0.5\% \) to \( 6.7\% \). We will use the midpoint of this range as the estimated proportion:

\[ p = \frac{0.005 + 0.067}{2} = \frac{0.072}{2} = 0.036 \]

Using the same formula for sample size:

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

Substituting the values:

\[ n = \frac{(1.96)^2 \cdot 0.036 \cdot (1 - 0.036)}{(0.02)^2} = \frac{(3.8416) \cdot 0.036 \cdot 0.964}{0.0004} \]

Calculating the numerator:

\[ 3.8416 \cdot 0.036 \cdot 0.964 \approx 0.135 \]

Thus,

\[ n = \frac{0.135}{0.0004} = 337.5 \]

Rounding up to the nearest whole number gives:

\[ \boxed{n = 334} \]

The comparison of the sample sizes calculated in Steps 1 and 2 shows a significant reduction in the required sample size when using a likely range for \( p \):

• Without assumptions: \( n = 2401 \)
• With assumptions: \( n = 334 \)

This indicates that using historical data to estimate the response rate can greatly reduce the sample size needed for accurate estimation.

Final Answer