Suppose the weight of lemons in a large crate has a distribution with mean 139 grams and standard deviation 4.2 grams. Consider the process of picking 3 lemons at random from the crate and putting them in a bag. Let *X* be the mean weight of the lemons in the bag. The standard deviation of *X* is

To find the standard deviation of the mean weight \( X \) of the lemons in the bag, we can use the concept of the standard deviation of the sample mean. When you take a sample of size \( n \) from a population, the standard deviation of the sample mean is given by:

\[
\text{Standard Deviation of } X = \frac{\sigma}{\sqrt{n}}
\]

Where:
- \( \sigma \) is the standard deviation of the population.
- \( n \) is the sample size.

In this case:
- The population standard deviation \( \sigma = 4.2 \) grams.
- The sample size \( n = 3 \).

So, the standard deviation of \( X \) (the mean weight of the lemons in the bag) is:

\[
\text{Standard Deviation of } X = \frac{4.2}{\sqrt{3}}
\]

Now let's calculate this:

\[
\text{Standard Deviation of } X = \frac{4.2}{\sqrt{3}} \approx \frac{4.2}{1.732} \approx 2.42 \text{ grams}
\]

Therefore, the standard deviation of \( X \), the mean weight of the 3 lemons in the bag, is approximately **2.42 grams**.

Do you have any questions or would you like further details on this?

Here are 5 related questions you might find interesting:

1. How would the standard deviation of \( X \) change if you picked 5 lemons instead of 3?
2. What is the probability that the mean weight of 3 lemons is more than 140 grams?
3. How does the Central Limit Theorem apply to the distribution of \( X \)?
4. What is the variance of the mean weight of the 3 lemons?
5. How does increasing the sample size affect the standard error of the mean?

**Tip:** When working with the mean of a sample, remember that the standard error decreases as the sample size increases, making the sample mean a more reliable estimate of the population mean.

Explore how to calculate the standard deviation of the mean weight of lemons in a bag when picking 3 lemons randomly from a crate. Learn the formula σ / √n and its application with a population standard deviation of 4.2 grams. Suitable for advanced high school students and learners interested in statistics.

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