Question

You want to construct a 95% confidence interval for the performance of a large population of mutual funds. Assume returns are independent across funds, and the standard deviation of fund returns is 9.8 %. If you want the width of your interval to be 2.3 %, what sample size must you collect?

Assume sample is large enough that the sample mean is normally distributed.

Enter answer as the smallest integer sample that will accomplish your objective.

The answer is 279.

Answer 1

α = 95%

Z_α/2 = 1.96

The intervals will be Mean - Z_α/2 ​​​​​​​x σ / n^1/2 and Mean + Z_{α​​​​​​​/2} ​​​​​​​x σ / n^1/2​​​​​​​

Hence, the width = 2 x Z_{α​​​​​​​/2} ​​​​​​​x σ / n^1/2​​​​​​​ = 2.3%

Hence, 2 x 1.96 x 9.8% / n^1/2 = 2.3%

Hence, n^1/2 = 2 x 1.96 x 9.8% / 2.3% = 16.70261

Hence, the smallest integer sample that will accomplish your objective = n = 16.70261² = 278.9771372 = 279