Final answer:
To establish a sample size that ensures a margin of error within 5 percentage points for a 90% confidence level with a sample proportion of 0.50, the formula n = (Z² × p' × (1 - p')) / E² is used. A Z-score of 1.645 is applied considering it's a 90% confidence interval. The company can then calculate the minimum number of customers to survey.
Step-by-step explanation:
To determine the required sample size for the Consideration marketing objective using customer survey data, we will apply a confidence interval for the proportion method. Considering the scenario where the sample proportion (p') is assumed to be 0.50, which is the maximized variance scenario (the most conservative assumption for binary data), we aim for a 90 percent confidence level with a margin of error of 5 percentage points. The sample size can be calculated using the formula: n = (Z² × p' × (1 - p')) / E², where Z is the Z-score corresponding to a 90% confidence level, and E is the desired margin of error.
To find the Z-score associated with a 90% confidence interval, we look up the Z-score that cuts off the top 5% of the normal distribution since this gives us the top 10% of the two-tailed distribution, yielding a Z-score of approximately 1.645. With p' = 0.50 and E = 0.05, the calculation would be: n = (1.645² × 0.50 × 0.50) / 0.05². After computing, we can determine the minimum number of customers the company needs to survey.