Seventy million pounds of trout are grown in the U.S. every year. Farm-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 34 farm-raised trout is selected. The mean fat content for the sample is 30.5 grams per pound. Find the probability of observing a sample mean of 30.5 grams of fat per pound or less in a random sample of 34 farm-raised trout.

Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.

To find the probability of observing a sample mean of 30.5 grams of fat per pound or less in a random sample of 34 farm-raised trout, we can use the Central Limit Theorem and the standard deviation of the sample mean.

First, let's calculate the standard deviation of the sample mean (also known as the standard error). The formula for the standard deviation of the sample mean is:

Standard error = standard deviation / √(sample size)

In this case, the standard deviation of the sample mean (standard error) can be calculated as:

Standard error = 7.5 / √34 ≈ 1.2880 (rounded to four decimal places)

Next, we need to calculate the z-score associated with the sample mean of 30.5 grams of fat per pound. The z-score represents how many standard deviations the observation is away from the mean. The z-score formula is:

z = (sample mean - population mean) / standard error

In this case, the population mean is the same as the sample mean, which is 30.5 grams of fat per pound. So, the z-score can be calculated as:

z = (30.5 - 32) / 1.2880 ≈ -1.4003 (rounded to four decimal places)

Now, we need to find the probability of observing a z-score of -1.4003 or less in a standard normal distribution. We can use a standard normal distribution table or a calculator to find this probability.

Looking up the z-score -1.4003 in a standard normal distribution table, we find that the probability is approximately 0.0808 (rounded to four decimal places).

Therefore, the probability of observing a sample mean of 30.5 grams of fat per pound or less in a random sample of 34 farm-raised trout is approximately 0.0808 (rounded to three decimal places).