A sample size of 25 is drawn from a normal population with a mean of 62. If the standard deviation from the sampling distribution is 3.5, what is the standard deviation of the orig... A sample size of 25 is drawn from a normal population with a mean of 62. If the standard deviation from the sampling distribution is 3.5, what is the standard deviation of the original population? Read more

Steps to Solve

1. Identify the given information

We know the following:

• Sample size ($n$) = 25
• Standard deviation of the sampling distribution ($\sigma_{\bar{x}}$) = 3.5

2. Use the formula for the standard deviation of the sampling distribution

The relationship between the standard deviation of the sampling distribution and the standard deviation of the original population is given by the formula:

$$ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} $$

Where:

• $\sigma_{\bar{x}}$ is the standard deviation of the sampling distribution
• $\sigma$ is the standard deviation of the original population
• $n$ is the sample size

3. Rearrange the formula to solve for the population standard deviation

To find the population standard deviation ($\sigma$), we rearrange the formula:

$$ \sigma = \sigma_{\bar{x}} \cdot \sqrt{n} $$

4. Plug in the values and calculate

Substituting the known values into the equation:

$$ \sigma = 3.5 \cdot \sqrt{25} $$

Calculating $\sqrt{25}$:

$$ \sqrt{25} = 5 $$

Thus,

$$ \sigma = 3.5 \cdot 5 $$

5. Perform the final calculation

Calculating ( \sigma ):

$$ \sigma = 17.5 $$