Question

A consumer products firm has recently introduced a new brand. The firm would like to estimate the proportion of people in its target market segment who are aware of the new brand. As part of a larger market research study, it was found that in a sample of 125 randomly selected individuals from the target market segment, 87 individuals were aware of the firm’s new brand.

1.Construct a 90% confidence interval for the proportion of individuals in the target market segment who are aware of the firm’s new brand. Use by hand calculations and R commands to build the confidence interval. For by hand calculations, use R to compute the z-multiple. Provide the step calculations and commands used in R. State the confidence interval in the two forms: point estimate ± margin of error and [lower limit, upper limit].

Answer 1

To construct a 90% confidence interval for the proportion of people in the target market segment who are aware of the new brand, calculate the point estimate and the margin of error using the z-multiple and standard error. Finally, construct the confidence interval using the point estimate and margin of error.

Step-by-step explanation:

To construct a 90% confidence interval for the proportion of people in the target market segment who are aware of the new brand, we first need to calculate the point estimate. The point estimate is the sample proportion, which can be calculated by dividing the number of individuals in the sample who are aware of the new brand by the total sample size:

Point Estimate = Number of individuals aware of the new brand / Total sample size = 87 / 125 = 0.696

The next step is to calculate the margin of error, which depends on the z-multiple and the standard error. The z-multiple is determined by the desired confidence level, and the standard error can be calculated using the formula:

Standard Error = sqrt((Point Estimate * (1 - Point Estimate)) / Sample size) = sqrt((0.696 * (1 - 0.696)) / 125) = 0.045

Since the desired confidence level is 90%, the z-multiple can be obtained from the Z-score table or using R commands. In R, the command to find the z-multiple is:

qnorm((1 + 0.90) / 2)

The result is approximately 1.645. Now we can calculate the margin of error:

Margin of Error = Z-multiple * Standard Error = 1.645 * 0.045 = 0.074

Finally, we can construct the confidence interval using the point estimate and the margin of error:

Confidence Interval = Point Estimate ± Margin of Error

Confidence Interval = 0.696 ± 0.074

Confidence Interval = [0.622, 0.770]