To determine the sample size ($n$) that results in a sampling distribution of the means with a standard deviation of 10, we need to use the formula for the standard error of the mean:

$\text{Standard Error of the Mean} = \frac{\sigma}{\sqrt{n}}$

Where:

• $\sigma$ is the population standard deviation (80 in this case).
• $n$ is the sample size.
• The standard error of the mean is the desired standard deviation for the sampling distribution of the means (10 in this case).

We can rearrange this equation to solve for $n$:

$10 = \frac{80}{\sqrt{n}}$

Now, solve for $n$:

1. Multiply both sides of the equation by $\sqrt{n}$: $10 \sqrt{n} = 80$

2. Divide both sides by 10: $\sqrt{n} = \frac{80}{10} = 8$

3. Square both sides to eliminate the square root: $n = 8^2 = 64$

Thus, the researcher should collect samples of 64 students to achieve a distribution of sample means with a standard deviation of 10.

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Would you like further explanation or additional details?

Here are 5 related questions for further exploration:

1. What is the relationship between sample size and standard error of the mean?
2. How does the central limit theorem apply to the sampling distribution of the mean?
3. How would the required sample size change if the desired standard deviation was 5 instead of 10?
4. What is the effect of increasing the population standard deviation on the sample size needed?
5. How does increasing the sample size affect the accuracy of estimating the population mean?

Tip: A larger sample size reduces the variability of the sample mean, making it a better estimate of the population mean.