Find an estimate of the sample size needed to obtain the specified margin of error for the 95% confidence interval.
A researcher wishes to estimate the mean amount of money spent per month on food by households in a certain neighborhood. She desires a margin of error of . Past studies suggest that a population standard deviation of is reasonable. Estimate the minimum sample size needed to estimate the population mean with the stated accuracy.
Well, to estimate the minimum sample size needed, we will need a little bit of mathematical calculation. However, I will try my best to make it as enjoyable as possible!
Let's take a moment to appreciate the researcher's desire for accuracy in estimating the mean amount of money spent per month on food. After all, who doesn't want accurate data when it comes to food?
Now, to calculate the sample size, we need to consider the desired margin of error and the known population standard deviation. The formula for estimating the sample size is:
n = (Z * σ / E)^2
Where:
n is the sample size we're looking for (just like looking for the last cookie in the jar),
Z is the z-score that corresponds to the desired confidence level (in this case, 95%, because who doesn't want to be 95% confident about things?),
σ is the population standard deviation (which the past studies suggest is reasonable, so let's trust that),
E is the desired margin of error (our accuracy goal).
So, using this formula, we plug in the values we have:
n = (Z * σ / E)^2
And there you go! You have a formula to estimate the minimum sample size needed. Just plug in the appropriate values and calculate. Remember, accuracy is important, but so is laughter! So, make sure to take a little break and laugh a bit before tackling those calculations. Good luck!
To estimate the required sample size, we can use the formula:
n = (Z * σ / E)^2
Where:
n = sample size
Z = Z-score corresponding to the desired level of confidence (in this case, 95% confidence level)
σ = population standard deviation
E = margin of error
Since the Z-score for a 95% confidence level is approximately 1.96 (assuming a large sample size), and the population standard deviation (σ) is already given, we can substitute these values into the formula:
n = (1.96 * σ / E)^2
Provide the value for the desired margin of error (E) to get the final estimate of the sample size needed.
To estimate the minimum sample size needed to estimate the population mean with a specified margin of error, we can use the following formula:
n = (Z * σ / E)^2
Where:
n = sample size
Z = Z-score corresponding to the desired confidence level (in this case, 95% or 1.96)
σ = population standard deviation
E = margin of error
In this case, the desired margin of error is denoted as and the population standard deviation as . Let's substitute the values into the formula:
n = (1.96 * σ / )^2
Now, we need to estimate the value of the population standard deviation (). The statement mentions that past studies suggest it is reasonable, but we do not have the actual value. Without the population standard deviation, we cannot find the exact sample size required.
If you have access to the data from past studies or any other relevant information that provides an estimate for the population standard deviation (), you can substitute that value into the formula to calculate the minimum sample size (n) needed.