Suppose the weight of bananas in a large crate has a distribution with mean 158 grams and standard deviation 19.2 grams. Consider the process of picking 8 bananas at random from the crate and putting them in a bag. Let X be the mean weight of the bananas in the bag. Assuming the weights of bananas are Normally distributed, the probability that the mean weight of the bananas in the bag is at least 174 grams is

To solve this problem, we'll use the concept of the sampling distribution of the sample mean \( X \). Given that the weights of the bananas are Normally distributed, the sample mean \( X \) will also be Normally distributed. Here’s a step-by-step breakdown:

### 1. Parameters of the Population
- Mean weight of bananas \( \mu = 158 \) grams
- Standard deviation of the weights \( \sigma = 19.2 \) grams

### 2. Parameters of the Sampling Distribution
For a sample size \( n = 8 \), the mean of the sampling distribution of \( X \) (the sample mean) is the same as the population mean:
\[
\mu_X = \mu = 158 \text{ grams}
\]
The standard deviation of the sampling distribution of \( X \) (standard error) is:
\[
\sigma_X = \frac{\sigma}{\sqrt{n}} = \frac{19.2}{\sqrt{8}} \approx 6.787 \text{ grams}
\]

### 3. Standardizing the Problem
We want to find the probability that the mean weight \( X \) of the bananas in the bag is at least 174 grams. This requires finding:
\[
P(X \geq 174)
\]
First, we standardize this using the Z-score formula:
\[
Z = \frac{X - \mu_X}{\sigma_X} = \frac{174 - 158}{6.787} \approx 2.36
\]

### 4. Finding the Probability
Now, we need to find the probability corresponding to \( Z \geq 2.36 \) using the standard normal distribution.

Using standard normal distribution tables or a calculator:
\[
P(Z \geq 2.36) \approx 1 - P(Z \leq 2.36) \approx 1 - 0.9909 = 0.0091
\]

### 5. Conclusion
The probability that the mean weight of the bananas in the bag is at least 174 grams is approximately **0.0091**, or 0.91%.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. What is the probability that the mean weight of 8 bananas is less than 150 grams?
2. How would the probability change if the sample size were increased to 16 bananas?
3. What is the probability that the mean weight of 8 bananas is exactly 158 grams?
4. How would the calculation change if the distribution of weights was not normal?
5. Can you find the 95% confidence interval for the mean weight of 8 bananas?

**Tip:** Remember that the standard deviation of the sample mean decreases as the sample size increases, making the distribution of the sample mean narrower.

Calculate the probability that the mean weight of 8 bananas picked randomly from a crate, with a known normal distribution (mean 158 grams, standard deviation 19.2 grams), is at least 174 grams. Understand the sampling distribution, Z-score calculation, and normal distribution tables.

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