A researcher has gathered data on the daily returns on a portfolio of call options over a recent 250-day period. The mean daily return has been 0.1%, and the sample standard deviation of daily portfolio returns is 0.25%. The researcher believes that the mean daily portfolio return is not equal to zero.

- Construct a 95% confidence interval for the population mean daily return over the 250-day sample period.

Answer:

- Given a sample size of 250 with a standard deviation of 0.25%, the standard error can be computed as .
- At the 5% level of significance, the critical z-values for the confidence interval are z0.025 = 1.96 and −z0.025 = –1.96. Thus, given a sample mean equal to 0.1%, the 95% confidence interval for the population mean is:

0.1 – 1.96(0.0158) ≤ µ ≤ 0.1 + 1.96(0.0158), or

0.069% ≤ µ ≤ 0.131%

Question:

In the second step, why do we use the z-statistic instead of the t-statistic?

The promblem doesn’t show population variance, and the standard error is calculated by using sample standdard deviation.