Question

The Blue Wild Fish Company sells a particular species of fish that live in a local lake. In the entire lake the distribution of fish lengths follows a normal distribution with mean 26 inches and standard deviation 7.8 inches.
A fisherman goes out and catches 45 fish every day. If we were to look at the distribution of the average lengths of the fish caught on each day we would be looking at a sampling distribution.
The mean of this sampling distribution would be ____ inches.

Answer 1

Answer:

Mean of the sampling distribution, $\bar{x} = \mu = 26 inches$

Step-by-step explanation:

The entire lake's fish population has a normal distribution of the lengths with mean, $\mu$ = 26 inches and standard deviation, $\sigma$ = 7.8 inches.

A fisherman catches 45 fish every day.

The 45 fish caught every day represent a sample of the fish population in the lake.

The average lengths of the fish that are caught every day would be a sampling distribution.

Since the population distribution of the lengths of the fish is a normal distribution we can say that the samples will also be normally distributed.

Since the samples are also normally distributed like the population then we can say that the mean of the sampling distribution will be the same as that of the population.

Hence mean of the given sampling distribution, $\bar{x} = \mu = 26 inches$