Question

The mean duration to harvest habanero peppers is 75 days with a standard deviation of 12 days. The farmer chooses a random sample of 25 habaneros. Use this information for problems a-b.

a) Find the standard error of the mean.
b) For a random sample of 25 habaneros, what interval centered on the mean of the sampling distribution captures 68% of the sample means?
So, for a random sample of 25 habanero peppers, 68% of the sample means fall between ___ and ___ days.

Answer 1

a) The standard error of the mean is 2.4 days. b) The interval centered on the mean that captures 68% of the sample means is 72.6 to 77.4 days.

Step-by-step explanation:

a) The standard error of the mean can be calculated using the formula: standard error = standard deviation / square root of sample size. In this case, the standard deviation is 12 days and the sample size is 25. Therefore, the standard error of the mean is 12 / √25 = 12 / 5 = 2.4 days.

b) To find the interval centered on the mean of the sampling distribution that captures 68% of the sample means, we need to find the z-score corresponding to 68% in a standard normal distribution. The z-score can be found using a z-table or a calculator, and it is approximately 1. Therefore, the interval can be calculated as follows:

Lower limit = mean - (z-score * standard error) = 75 - (1 * 2.4) = 72.6 days

Upper limit = mean + (z-score * standard error) = 75 + (1 * 2.4) = 77.4 days