Let x represents a random number of apples a tree in Oak Glen, California produces in a given season. It has a mean of 101 apples and standard deviation of 50 apples, then the standard deviation of the sampling distribution of the means with sample size 100 is 5. True O False

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**Question:**

Let \( x \) represent a random number of apples a tree in Oak Glen, California produces in a given season. It has a mean of 101 apples and a standard deviation of 50 apples. Then, the standard deviation of the sampling distribution of the means with a sample size of 100 is 5.

- ○ True
- ○ False

**Explanation:**

To determine if the statement is true, recall that the standard deviation of the sampling distribution of the sample mean (also known as the standard error) is calculated using the formula:

\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\]

where:
- \(\sigma_{\bar{x}}\) is the standard deviation of the sample mean,
- \(\sigma\) is the population standard deviation (50 apples in this case),
- \(n\) is the sample size (100 in this case).

Calculating:

\[
\sigma_{\bar{x}} = \frac{50}{\sqrt{100}} = \frac{50}{10} = 5
\]

Therefore, the statement is **True**.
Transcribed Image Text:**Question:** Let \( x \) represent a random number of apples a tree in Oak Glen, California produces in a given season. It has a mean of 101 apples and a standard deviation of 50 apples. Then, the standard deviation of the sampling distribution of the means with a sample size of 100 is 5. - ○ True - ○ False **Explanation:** To determine if the statement is true, recall that the standard deviation of the sampling distribution of the sample mean (also known as the standard error) is calculated using the formula: \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \] where: - \(\sigma_{\bar{x}}\) is the standard deviation of the sample mean, - \(\sigma\) is the population standard deviation (50 apples in this case), - \(n\) is the sample size (100 in this case). Calculating: \[ \sigma_{\bar{x}} = \frac{50}{\sqrt{100}} = \frac{50}{10} = 5 \] Therefore, the statement is **True**.
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