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?
Understand the Problem
The question asks for the calculation of the standard deviation of the original population, given a sample size and the standard deviation of the sampling distribution. To solve it, we can use the formula for the standard deviation of the population, which relates sample size and standard deviation of the sampling distribution.
Answer
The standard deviation of the original population is $\sigma = 17.5$.
Answer for screen readers
The standard deviation of the original population is $\sigma = 17.5$.
Steps to Solve
- Understand the relationship between standard deviations
To find the standard deviation of the original population ($\sigma$), we can use the formula for the standard deviation of the sampling distribution, which is given by: $$ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} $$ where $\sigma_{\bar{x}}$ is the standard deviation of the sampling distribution, $\sigma$ is the standard deviation of the original population, and $n$ is the sample size.
- Substitute the known values into the formula
From the problem, we know:
- Sample size, $n = 25$
- Standard deviation of the sampling distribution, $\sigma_{\bar{x}} = 3.5$
We can substitute these values into the formula: $$ 3.5 = \frac{\sigma}{\sqrt{25}} $$
- Solve for the population standard deviation
Now, we need to rewrite the formula to solve for $\sigma$: $$ \sigma = 3.5 \times \sqrt{25} $$
Now calculate $\sqrt{25}$: $$ \sqrt{25} = 5 $$
Then substitute back into the equation: $$ \sigma = 3.5 \times 5 $$
- Perform the final multiplication
Now multiply: $$ \sigma = 17.5 $$
The standard deviation of the original population is $\sigma = 17.5$.
More Information
This problem illustrates the relationship between the sample standard deviation and the population standard deviation using the Central Limit Theorem. The standard deviation of the sampling distribution is often less than the population standard deviation due to the size of the samples taken.
Tips
- Misunderstanding the relationship: Confusing the standard deviation of the sampling distribution with the population standard deviation. It's crucial to follow the relationship outlined in the formula.
- Not taking the square root: Forgetting to properly calculate the square root of the sample size when using the formula.
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