Question

Seventy million pounds of trout are grown in the U.S. every year. Far-raised trout contain an average of 32 grams of fat per pound, with a standard deviation of 7.5 grams of fat per pound. A random sample of 36 farm-raised trout is selected. The mean fat content for the sample is 31.8 grams per pound. Find the probability of observing a sample mean of 31.8 grams of fat per pound or less in a random sample of 36 farm-raised trout. Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.

Answer 1

Solution:

Given:

Mean = = 32 grams

Standard deviation =    = 7.5 grams

Sample size = n = 36

The mean fat content for the sample is 31.8 grams per pound.

Sample mean = 31.8

We have to find:

P( sample mean of 31.8 grams of fat per pound or less in a random sample of 36 farm-raised trout. ) =..........?

Since sample size n = 36 is large , we can use Central limit theorem which states that for large sample size n ,
sampling distribution of sample mean is approximately normal with mean of sample means:

and standard deviation of sample means is:

Find z score :

thus

Look in z table for z = -0.1 and 0.06 and find corresponding area.

P( Z< -0.16) = 0.4364

thus