All the Fish in the Sea

Smaller fish are much more common than larger fish. This is reflected in the distribution of fish. It looks like this (definitely not a bell curve!):

The population of fish has the following parameters.

• $\mu = 9$ lbs.
• $\sigma = 9$ lbs.

If I catch a sample of 20 fish, randomly selected, my sample will probably reflect the true population distribution. Here are the weights (in lbs.) of the fish I caught:

0.69, 1.53, 2.07, 2.17, 2.31, 2.70, 3.03, 4.25, 5.51, 5.99, 6.96, 8.38, 9.18, 10.32, 12.12, 12.74, 14.39, 20.40, 21.11, 22.27

My sample of fish has the following statistics.

• $\bar{x} = 8.4$ lbs.
• $s = 6.8$ lbs.

Questions

1. What is the 95% confidence interval for $\mu$?

2. Does the confidence interval contain the true population mean?

3. Why can we use a tool (confidence intervals) that relies on bell curves to analyze a population that definitely does not have a bell shaped distribution?