The problem shown in the image involves finding the probability related to a sample mean, which can be solved using the z-score formula for a sampling distribution.

Here is a breakdown of the given information:

• Population mean $\mu = 32$ grams of fat per pound.
• Population standard deviation $\sigma = 7.5$ grams of fat per pound.
• Sample size $n = 35$.
• Sample mean $\bar{x} = 31.5$ grams of fat per pound.

Step 1: Calculate the standard error of the mean (SEM)

The standard error of the mean is given by: $SEM = \frac{\sigma}{\sqrt{n}}$ Substitute the given values: $SEM = \frac{7.5}{\sqrt{35}}$

Step 2: Calculate the z-score

The z-score formula for a sample mean is: $z = \frac{\bar{x} - \mu}{SEM}$ Substitute the values: $z = \frac{31.5 - 32}{SEM}$

Step 3: Use the z-score to find the probability

Once the z-score is calculated, we can use the standard normal distribution to find the probability of observing a sample mean of 31.5 grams of fat per pound or less.

Let me calculate these values for you.The standard error of the mean (SEM) is approximately 1.2677, and the z-score is approximately $-0.3944$.

Step 3: Find the probability

Now, to find the probability of observing a sample mean of 31.5 grams or less, we need to look up the cumulative probability associated with a z-score of $-0.3944$ in the standard normal distribution.

Let me calculate this probability for you.The probability of observing a sample mean of 31.5 grams of fat per pound or less in a random sample of 35 farm-raised trout is approximately 0.3466 or 34.66%.

Would you like more details on any step, or do you have any other questions?

Here are 5 related questions to explore further:

1. How would the probability change if the sample size increased?
2. What is the interpretation of the z-score in this context?
3. How does the standard deviation impact the shape of the sampling distribution?
4. How would you calculate the probability for a sample mean greater than 31.5 grams per pound?
5. How is the Central Limit Theorem applied in this problem?

Tip: The standard normal distribution table can be used to quickly find probabilities for any z-score.