Question

Product Response Rate. 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 advanced 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

a. Without making any assumptions, determine the sample size required.

b. Historically, product response rates for products sold by this company have ranged from $0.5\%$to $4.9\%$. 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.

c. Compare the results from parts (a) and (b).

d. Discuss the possible consequences if the assumption made in part (b) turns out to be incorrect.

Short answer

(a) The sample size is$9604$.

(b) The sample size is$1984$.

(c) From part (b), the sample size required to assume product response rate is between $0.5\%$to $4.9\%$is $n=1984.$.

(d) If this assumption is inaccurate, it will have a significant influence on the company's revenue. Since the manufacturing order will be determined based on this information, if more units are made than are required, the product will go unsold. In addition, if fewer units are produced than are required, there would be a shortage of merchandise on the market.

Step-by-step solution

Part (a) Step 1: Given information

Director of market research wants to be $95\%$ confident that the product response rate would be accurate to within $1\%$.

Part (a) Step 2: Explanation

A $(1-\alpha )$level confidence interval for a population that has margin of error of at most $E$, sample size is obtained by

role="math" localid="1651488828571" $n=0.25\times {\left(\frac{z_a/2}{E}\right)}^2$

When, $\alpha =0.05$, confidence interval is $95\%$

$z_a/2=z_{0.025}$

$=1.96$

Here accuracy of product response rate need to be within $1\%$.

So, $E=0.01$, Therefore, the required sample size $n$is calculated as

$n=0.25{\left(\frac{1.96}{0.01}\right)}^2$

$n=9604.$

Part (b) Step 1: Given information

Director of market research wants to be $95\%$ confident that the product response rate would be accurate to within $1\%$.

Part (b) Step 2: Explanation

A $(1-\alpha )$level confidence interval for a population that has margin of error of at most $E$, sample size is obtained by

$n=0.25\times {\left(\frac{z_{a}2}{E}\right)}^2$

When $\alpha =0.05$, the confidence interval is $95\%$

$z_{a/2}=z_{0.025}$

$=1.96$

Here product response rate is between $0.005$to $0.049$.

The margin of error is equal to the width of the confidence interval divided by $2$.

So, $E=0.022$.

Therefore, the required sample size $n$is calculated as

$n=0.25{\left(\frac{1.96}{0.022}\right)}^2$

$n=1984$.

Part (c) Step 1: Given information

Director of market research wants to be $95\%$ confident that the product response rate would be accurate to within $1\%$.

Part (c) Step 2: Explanation

From part (a), the sample size required to restrict margin of error at most $0.01$is $n=9604$.

From part (b), the sample size required to assume product response rate is between $0.5\%$to $4.9\%$is$n=1984$.

Part (d) Step 1: Given information

Director of market research wants to be $95\%$ confident that the product response rate would be accurate to within $1\%$.

Part (d) Step 2: Explanation

The director of market research wants to be able to be$95\%$ certain that the product response rate is accurate to within $1\%$. She also assumes that the product response rate for the new product line will be in the range of $0.5\%$to $4.9\%$, based on prior product response rate data.

If this assumption is inaccurate, it will have a significant influence on the company's revenue. Since the manufacturing order will be determined based on this information, if more units are made than are required, the product will go unsold. In addition, if fewer units are produced than are required, there would be a shortage of merchandise on the market.