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Answer :
To solve this problem, we're looking for the standard deviation of the average SAT math scores for a Simple Random Sample (SRS) of 100 students. This is a problem involving the sampling distribution of the sample mean.
Here's how we can find the solution step-by-step:
1. Understand the Central Limit Theorem:
- The Central Limit Theorem tells us that if you have a population with a mean (µ) and standard deviation (σ), and you take sufficiently large random samples of size n, then the distribution of the sample mean will be approximately normally distributed.
2. Mean and Standard Deviation of the Sample Mean:
- The mean of the sample mean is the same as the mean of the population.
- The standard deviation of the sample mean (often referred to as the standard error) is calculated using the formula:
[tex]\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}}
\][/tex]
where [tex]\(\sigma\)[/tex] is the population standard deviation, and [tex]\(n\)[/tex] is the sample size.
3. Plug in the Given Values:
- For this problem, the population standard deviation ([tex]\(\sigma\)[/tex]) is 114.
- The sample size ([tex]\(n\)[/tex]) is 100 students.
4. Calculate the Standard Error:
- Substitute the values into the formula:
[tex]\[
\text{Standard Error} = \frac{114}{\sqrt{100}}
\][/tex]
- Since the square root of 100 is 10, we have:
[tex]\[
\frac{114}{10} = 11.4
\][/tex]
So, the standard deviation of the average scores you get when choosing an SRS of 100 students is approximately 11.4. The correct answer is (c) 114 / √100 = 11.4.
Here's how we can find the solution step-by-step:
1. Understand the Central Limit Theorem:
- The Central Limit Theorem tells us that if you have a population with a mean (µ) and standard deviation (σ), and you take sufficiently large random samples of size n, then the distribution of the sample mean will be approximately normally distributed.
2. Mean and Standard Deviation of the Sample Mean:
- The mean of the sample mean is the same as the mean of the population.
- The standard deviation of the sample mean (often referred to as the standard error) is calculated using the formula:
[tex]\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}}
\][/tex]
where [tex]\(\sigma\)[/tex] is the population standard deviation, and [tex]\(n\)[/tex] is the sample size.
3. Plug in the Given Values:
- For this problem, the population standard deviation ([tex]\(\sigma\)[/tex]) is 114.
- The sample size ([tex]\(n\)[/tex]) is 100 students.
4. Calculate the Standard Error:
- Substitute the values into the formula:
[tex]\[
\text{Standard Error} = \frac{114}{\sqrt{100}}
\][/tex]
- Since the square root of 100 is 10, we have:
[tex]\[
\frac{114}{10} = 11.4
\][/tex]
So, the standard deviation of the average scores you get when choosing an SRS of 100 students is approximately 11.4. The correct answer is (c) 114 / √100 = 11.4.
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