Suppose the weight of tomatoes in a large crate has a distribution with mean 141 grams and standard deviation 25.3 grams. Consider the process of picking 6 tomatoes at random from the crate and putting them in a bag. Let X be the mean weight of the tomatoes in the bag. Assuming the weights of tomatoes are Normally distributed, the probability that the mean weight of the tomatoes in the bag is at least 164 grams is

Given that the weight of the tomatoes follows a normal distribution with a mean (\(\mu\)) of 141 grams and a standard deviation (\(\sigma\)) of 25.3 grams, we need to find the probability that the mean weight \(X\) of the 6 tomatoes in the bag is at least 164 grams.

### Step 1: Distribution of Sample Mean

The sample mean \(\overline{X}\) of the weights of the 6 tomatoes will also follow a normal distribution, but with a different mean and standard deviation:

- The mean of \(\overline{X}\) will be the same as the population mean:
  \[
  \mu_{\overline{X}} = 141 \text{ grams}
  \]

- The standard deviation of \(\overline{X}\), also known as the standard error (SE), is calculated as:
  \[
  \sigma_{\overline{X}} = \frac{\sigma}{\sqrt{n}} = \frac{25.3}{\sqrt{6}} \approx 10.327 \text{ grams}
  \]

So, \(\overline{X} \sim N(141, 10.327^2)\).

### Step 2: Convert to Standard Normal Distribution

We want to find the probability that \(\overline{X} \geq 164\) grams. This can be calculated by converting \(\overline{X}\) to a standard normal variable \(Z\):

\[
Z = \frac{\overline{X} - \mu_{\overline{X}}}{\sigma_{\overline{X}}}
\]

Substituting the values:

\[
Z = \frac{164 - 141}{10.327} \approx \frac{23}{10.327} \approx 2.227
\]

### Step 3: Find the Probability

We now need to find the probability \(P(Z \geq 2.227)\).

Using the standard normal distribution table (or a calculator), we find:

\[
P(Z \geq 2.227) \approx 1 - P(Z \leq 2.227) \approx 1 - 0.9871 = 0.0129
\]

### Conclusion

The probability that the mean weight of the tomatoes in the bag is at least 164 grams is approximately **0.0129**, or 1.29%.

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Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How does increasing the number of tomatoes affect the probability calculation?
2. What is the probability that the mean weight is exactly 141 grams?
3. How would the probability change if the standard deviation were larger?
4. What is the probability that the mean weight of tomatoes is less than 130 grams?
5. Can this method be used for non-normal distributions?

**Tip:** When working with sample means, always remember to divide the population standard deviation by the square root of the sample size to get the standard error.

Calculate the probability that the mean weight of tomatoes in a bag, sampled from a crate with a normal distribution of weights (mean 141 grams, standard deviation 25.3 grams), is at least 164 grams. Detailed step-by-step solution using normal distribution properties and sampling theory.

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