A normal population has mean 100 and variance 25. How large must the random sample
be if we want the standard error of the sample average to be 1.5?
I know the answer is 12. Would someone please be able to explain! and share what formula they used
Variance=25
Std Deviation = 5
so
1.5 = (5/sqrt(n)
n=11.11
then round up and
n=12
so i found the equation SE = sigma/sqrt(n) I plugged in 1.5 for SE and 5 for sigma. i got n= 11.1 but that is not the correct answer..
Well, I'm glad you asked! Let's dive into it.
To determine the sample size, we need to use the formula for the standard error of the sample mean (SE):
SE = sqrt(variance / sample size)
In this case, the variance is given as 25, and we want the standard error (SE) to be 1.5. Plugging in these values into the formula:
1.5 = sqrt(25 / sample size)
To solve for the sample size, we need to isolate it. Squaring both sides of the equation:
(1.5)^2 = (25 / sample size)
2.25 = 25 / sample size
Now, we can solve for the sample size by rearranging the equation:
sample size = 25 / 2.25
sample size ≈ 11.11
Now, keep in mind that sample size must be a whole number, so we need to round it up to the nearest integer. In this case, rounding it up gives us a sample size of 12, which matches the answer you mentioned earlier.
So, the minimum sample size required to achieve a standard error of 1.5 is 12.
To determine the sample size required to achieve a desired standard error of the sample average, we can use the formula:
Sample Size = (Z * σ) / SE
Where:
- Z is the Z-score corresponding to the desired confidence level (e.g., for a 95% confidence level, we use a Z-score of 1.96),
- σ is the population standard deviation, and
- SE is the desired standard error of the sample average.
In this case, we have the following information:
- Z = 1.96 (assuming a 95% confidence level)
- σ (population standard deviation) = √variance = √25 = 5
- SE (standard error of the sample average) = 1.5
Plugging these values into the formula, we get:
Sample Size = (1.96 * 5) / 1.5 = 6.53333
Since we cannot have a fraction of a sample, we round up the result to the nearest whole number. Therefore, the sample size required is 7, not 12.
It appears there might be a mistake in the given solution of 12.
To find the sample size, we can use the formula for the standard error of the sample average:
Standard Error (SE) = Standard Deviation (σ) / √(Sample Size)
Given that the population mean (μ) is 100 and the variance (σ^2) is 25, we can calculate the standard deviation (σ) as the square root of the variance, which is σ = √(25) = 5.
Now, we can substitute the values into the formula and solve for the sample size:
1.5 = 5 / √(Sample Size)
To isolate the sample size, we multiply both sides of the equation by √(Sample Size):
1.5 * √(Sample Size) = 5
Next, square both sides of the equation to get rid of the square root:
(1.5 * √(Sample Size))^2 = 5^2
Simplifying further:
2.25 * Sample Size = 25
Divide both sides of the equation by 2.25 to solve for the sample size:
Sample Size = 25 / 2.25
Sample Size ≈ 11.11
Since the sample size should be a whole number, we round up the value to the nearest integer, which gives us the final answer:
Sample Size ≈ 12
Therefore, to achieve a standard error of 1.5 for the sample average, the sample size should be 12.