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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.

Find 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, given that the population mean is 32 grams of fat per pound and the population standard deviation is 7.5 grams of fat per pound.

Calculate the probability of observing a sample mean of 31.5g fat per pound or less from a sample of 35 farm-raised trout. This problem explores sampling distribution, z-scores, and the standard normal distribution. Suitable for high school students studying probability and statistics.

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