Question

A major car magazine wants to estimate the proportion of customers who prefer to purchase domestically made cars. In a sample of 900 customers, 315 said they prefer to purchase domestically made cars. Construct a 95% confidence interval for the proportion of customers who prefer to purchase domestically made cars.

enter the upper limit for the confidence interval. (Enter your answer as a percent, rounded to 2 decimal places and include the % sign.)

enter the lower limit for the confidence interval. (Enter your answer as a percent, rounded to 2 decimal places and include the % sign.)

Is it reasonable for the magazine to claim that the proportion of customers who prefer to purchase domestically made cars is 40%? Enter "yes" or "no"

Answer 1

To construct a confidence interval for the proportion, we can use the formula: CI = p-hat ± z * sqrt((p-hat * (1 - p-hat)) / n), where p-hat is the observed proportion, z is the z-score corresponding to the desired confidence level, and n is the sample size. Calculating the upper and lower limits of the confidence interval, the 95% confidence interval for the proportion of customers who prefer to purchase domestically made cars is 30.6% to 39.4%. It is not reasonable for the magazine to claim that the proportion is 40%, as this value falls outside the confidence interval.

Step-by-step explanation:

To construct a confidence interval for the proportion, we can use the formula:

CI = p-hat ± z * sqrt((p-hat * (1 - p-hat)) / n)

where p-hat is the observed proportion, z is the z-score corresponding to the desired confidence level, and n is the sample size. In this case, p-hat = 315/900 = 0.35, z = 1.96 for a 95% confidence level, and n = 900. Plugging these values into the formula, we have:

CI = 0.35 ± 1.96 * sqrt((0.35 * (1 - 0.35)) / 900)

Calculating the upper and lower limits of the confidence interval:

Upper limit = 0.35 + 1.96 * sqrt((0.35 * (1 - 0.35)) / 900) = 0.394

Lower limit = 0.35 - 1.96 * sqrt((0.35 * (1 - 0.35)) / 900) = 0.306

Therefore, the 95% confidence interval for the proportion of customers who prefer to purchase domestically made cars is 30.6% to 39.4%. It is not reasonable for the magazine to claim that the proportion is 40%, as this value falls outside the confidence interval.