A statistics practitioner would like to estimate a population mean to within 50 units with 99% confidence given that the population standard deviation is 250.
a) What sample size should be used.
b) Repeat part (a) with a standard deviation of 50.
c) Repeat part (a) using a 95% confidence level.
d) Repeat part (a) when we wish to estimate the population mean to within 10 units.
Formula:
n = [(z-value * sd)/E]^2
...where n = sample size, use a z-table to represent the 99% confidence interval, sd = 250, E = 50, ^2 means squared, and * means to multiply.
Plug the values into the formula and finish the calculation. Round your answer to the next highest whole number.
For b), use 50 for sd.
For c), use 95% interval.
For d), use 10 for E.
I hope this will help get you started.
a)166
b)7
c)97
d)4148
To calculate the sample size needed to estimate the population mean with a given level of confidence and margin of error, we can use the formula:
n = (Z * σ / E)^2
where:
n = sample size
Z = Z-score corresponding to the desired level of confidence
σ = population standard deviation
E = desired margin of error (half the width of the confidence interval)
a) Given that the population standard deviation is 250, the desired margin of error is 50, and the desired confidence level is 99%, we can calculate the sample size as follows:
Z = Z-score for a 99% confidence level is 2.576
n = (2.576 * 250 / 50)^2
n ≈ 330.66
Therefore, a sample size of at least 331 should be used.
b) If the standard deviation is 50 and all other values remain the same, the calculation would be:
Z = 2.576
E = 50
n = (2.576 * 50 / 50)^2
n ≈ 26.06
Therefore, a sample size of at least 27 should be used.
c) If the confidence level is 95% and all other values remain the same, the calculation would be:
Z = Z-score for a 95% confidence level is 1.96
n = (1.96 * 250 / 50)^2
n ≈ 96.04
Therefore, a sample size of at least 97 should be used.
d) If we want to estimate the population mean to within 10 units, the calculation would be:
Z = 2.576
E = 10
n = (2.576 * 250 / 10)^2
n ≈ 6656.9
Therefore, a sample size of at least 6657 should be used.
To calculate the sample size needed to estimate a population mean with a certain level of confidence and a given population standard deviation, you can use the formula:
n = (Z * σ / E)^2
Where:
n = sample size
Z = Z-score corresponding to the desired confidence level
σ = population standard deviation
E = desired margin of error (half the width of the confidence interval)
Let's calculate the sample size for each part of the question:
a) Given a population standard deviation of 250 and a desired margin of error of 50 units with a 99% confidence level, the Z-score corresponding to a 99% confidence level is approximately 2.576 (obtained from a standard normal distribution table). Plugging these values into the formula:
n = (2.576 * 250 / 50)^2
n ≈ 326 (rounded up to the nearest whole number)
So, a sample size of at least 326 should be used.
b) To repeat part (a) with a population standard deviation of 50, we can simply substitute the new standard deviation value into the formula:
n = (2.576 * 50 / 50)^2
n = 2.576^2
n ≈ 6.65
Since the sample size must be a whole number, we round up to the nearest whole number:
n = 7
So, a sample size of at least 7 should be used.
c) To repeat part (a) using a 95% confidence level, we need to find the Z-score corresponding to a 95% confidence level. From a standard normal distribution table, the Z-score for a 95% confidence level is approximately 1.96:
n = (1.96 * 250 / 50)^2
n ≈ 121
So, a sample size of at least 121 should be used.
d) To repeat part (a) with a desired margin of error of 10 units, we substitute the new margin of error into the formula:
n = (2.576 * 250 / 10)^2
n = 2.576^2 * 250^2 / 10^2
n ≈ 33347
So, a very large sample size of at least 33347 should be used to estimate the population mean to within 10 units.