A marketing class is engaged in a project to estimate the proportion of homes receiving both HBO and Showtime cable service. The class plans to use a confidence coefficient of 0.95 and a maximum error of the estimate equal to 0.10. Determine the sample size for the class to use.

To determine the sample size needed for this project, we need to use the formula for sample size calculation:

n = (Z^2 * p * (1 - p)) / E^2

Where:
n = sample size
Z = z-score corresponding to the desired confidence level (confidence coefficient)
p = estimated proportion of homes receiving both HBO and Showtime cable service (this is typically unknown and needs to be estimated)
E = maximum error or margin of error

Here, the confidence coefficient is 0.95, and the maximum error is 0.10.

To calculate the sample size, we need to estimate the proportion of homes receiving both HBO and Showtime cable service (p). Since we don't have an estimate, we'll assume a conservative value of 0.5, which maximizes the sample size requirement.

Now, let's calculate the sample size:

n = (Z^2 * p * (1 - p)) / E^2
= (1.96^2 * 0.5 * (1 - 0.5)) / 0.10^2
≈ 384.16

Since we can't have a fraction of a sample, we round the result up to the nearest whole number.

Therefore, the sample size that the marketing class should use is approximately 385.