Final answer:
To estimate the required sample size for a survey with a specified margin of error and confidence level, a statistical formula is used with an assumed proportion and the Z-value for the desired confidence. The closest given option to the calculated sample size would be the correct choice for the sample size needed.
Step-by-step explanation:
The question is related to determining the sample size necessary for a survey to estimate the proportion of computers that will use a new operating system with a certain degree of confidence and margin of error. To find the required sample size for estimating a population proportion with a given level of confidence and margin of error, the following formula is typically used:
n = (Z^2 * p * (1-p)) / E^2
Where:
- n is the sample size,
- Z is the Z-value associated with the desired level of confidence,
- p is the estimated proportion (if unknown, 0.5 is used as it maximizes the sample size),
- E is the margin of error.
For a 97% confidence level, which corresponds to a Z value of approximately 2.17, with a margin of error E of 0.015 (1.5 percentage points), the formula would be inserted with these values.
Since an exact value is not provided for the proportion p, the conservative approach of using p = 0.5 is applied as it provides the maximum sample size that might be needed.
By applying the formula, we can determine which of the provided options (385, 500, 650, or 800) is adequate for the programmer's requirements.