Answer :
To solve the problem of finding the standard deviation of the average scores from a sample of 100 students' SAT math scores, we can follow these steps:
1. Understand the concept of sampling distribution of the sample mean:
- When we take a simple random sample (SRS) of size [tex]\( n \)[/tex] from a population with a known mean and standard deviation, the mean of the sample will also have a distribution. The mean of this sampling distribution will be the same as the population mean, and its standard deviation, known as the standard error, will be smaller than the population standard deviation.
2. The formula for the standard error:
- The standard error (SE) of the sample mean is given by the formula:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
where [tex]\( \sigma \)[/tex] is the population standard deviation, and [tex]\( n \)[/tex] is the sample size.
3. Apply the given values to the formula:
- In this problem, the population standard deviation ([tex]\( \sigma \)[/tex]) is 114, and the sample size ([tex]\( n \)[/tex]) is 100.
4. Calculate the standard error:
- Using the formula for the standard error, substitute the given values:
[tex]\[
SE = \frac{114}{\sqrt{100}}
\][/tex]
- Since [tex]\(\sqrt{100} = 10\)[/tex], the calculation becomes:
[tex]\[
SE = \frac{114}{10} = 11.4
\][/tex]
5. Choose the correct option:
- Based on the calculated standard error of 11.4, the correct answer is option (c): [tex]\( 114 / \sqrt{100} = 11.4 \)[/tex].
Therefore, the standard deviation of the average scores you get, also known as the standard error, is 11.4.
1. Understand the concept of sampling distribution of the sample mean:
- When we take a simple random sample (SRS) of size [tex]\( n \)[/tex] from a population with a known mean and standard deviation, the mean of the sample will also have a distribution. The mean of this sampling distribution will be the same as the population mean, and its standard deviation, known as the standard error, will be smaller than the population standard deviation.
2. The formula for the standard error:
- The standard error (SE) of the sample mean is given by the formula:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
where [tex]\( \sigma \)[/tex] is the population standard deviation, and [tex]\( n \)[/tex] is the sample size.
3. Apply the given values to the formula:
- In this problem, the population standard deviation ([tex]\( \sigma \)[/tex]) is 114, and the sample size ([tex]\( n \)[/tex]) is 100.
4. Calculate the standard error:
- Using the formula for the standard error, substitute the given values:
[tex]\[
SE = \frac{114}{\sqrt{100}}
\][/tex]
- Since [tex]\(\sqrt{100} = 10\)[/tex], the calculation becomes:
[tex]\[
SE = \frac{114}{10} = 11.4
\][/tex]
5. Choose the correct option:
- Based on the calculated standard error of 11.4, the correct answer is option (c): [tex]\( 114 / \sqrt{100} = 11.4 \)[/tex].
Therefore, the standard deviation of the average scores you get, also known as the standard error, is 11.4.
