Let's go through each part of your problem step by step.

(a) Why is the sampling distribution of $\bar{x}$ approximately normal?

The correct answer here is:

D. The sampling distribution of $\bar{x}$ is approximately normal because the sample size is large enough.

This is due to the Central Limit Theorem (CLT), which states that for sufficiently large sample sizes (typically $n \geq 30$), the sampling distribution of the sample mean will be approximately normal, regardless of the population's original distribution. Since the sample size is 50, it meets this criterion.

(b) What is the mean and standard deviation of the sampling distribution of $\bar{x}$?

For this part:

• The mean of the sampling distribution of $\bar{x}$ is the same as the population mean $\mu = 3$.

• The standard deviation of the sampling distribution of $\bar{x}$ (also called the standard error) is calculated using the formula: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ where $\sigma = \sqrt{3}$ (as given) and $n = 50$ (sample size).

  $\sigma_{\bar{x}} = \frac{\sqrt{3}}{\sqrt{50}} = \frac{1.732}{7.071} \approx 0.245$

Thus, the mean is 3 and the standard deviation is approximately 0.245.

(c) What is the probability a simple random sample of 50 ten-gram portions of the food item results in a mean of at least 3.7 insect fragments?

To find $P(\bar{x} \geq 3.7)$, we can standardize this using the z-score formula: $z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}}$ Substituting the values:

$z = \frac{3.7 - 3}{0.245} \approx \frac{0.7}{0.245} \approx 2.857$

Now, using the standard normal distribution table or a calculator, the probability of a z-score of 2.857 or higher is approximately 0.0021. Thus: $P(\bar{x} \geq 3.7) \approx 0.0021$

Is this result unusual?

To determine if this result is unusual, we compare the probability with a typical threshold (commonly, probabilities less than 0.05 or 5% are considered unusual). Since $P = 0.0021$ is much smaller than 0.05, this event is unusual.

The correct answer is: D. This result is unusual because its probability is small.

---

Let me know if you need further clarifications! Here are some related questions to consider:

1. What is the Central Limit Theorem and why is it important?
2. How is the standard error related to sample size?
3. What factors affect the shape of the sampling distribution of $\bar{x}$?
4. How would changing the sample size impact the probability found in part (c)?
5. What would be the z-score if the sample mean were 3.2 instead of 3.7?

Tip: When calculating probabilities from a normal distribution, always standardize the values using z-scores.