A large bank believes that one-third of its checking customers have used at least one of the bank's other services during the past year. How large a sample is required to estimate the actual proportion within a range of plus and minus 0.04? Use the 98 percent level of confidence.
To determine the required sample size for estimating the actual proportion, we can use the formula for sample size calculation for proportions. The formula is:
n = (Z^2 * p * (1-p)) / E^2
Where:
- n is the required sample size
- Z is the z-score associated with the desired level of confidence
- p is the estimated proportion
- E is the margin of error (plus or minus value)
In this case, the bank wants to estimate the actual proportion within a range of plus and minus 0.04, which means the margin of error (E) is 0.04. The level of confidence is 98 percent, which corresponds to a z-score of 2.33 (from a standard normal distribution table).
Now we need to estimate the proportion (p). The bank believes that one-third (1/3) of its checking customers have used another service, so we can estimate p as 1/3 or 0.33.
Plugging these values into the formula:
n = (2.33^2 * 0.33 * (1-0.33)) / 0.04^2
Simplifying the calculation:
n = (5.4289 * 0.33 * 0.67) / 0.0016
n = 0.11967 / 0.0016
n ≈ 74.79
Therefore, a sample size of approximately 74.79 is required to estimate the actual proportion within a range of plus and minus 0.04, with a 98 percent level of confidence. Since we cannot have a fraction of a person in a sample, we round up the result to the nearest whole number.
Therefore, the bank would need a sample size of at least 75 customers to meet these criteria.