a researcher is interested in studying people's mean age in a certain region. If the population standard deviation is known to be 8 years and 1.5 year of error margin is allowed, find the minimum simple size the researcher needs to use, knowing that he is going to conduct his study using 95% confidence level.
To calculate the minimum sample size needed for this study, we can use the formula for sample size estimation for estimating the population mean with a known population standard deviation.
The formula is:
n = (Z * σ / E)²
Where:
n = sample size
Z = Z-score for the desired confidence level (95% confidence level corresponds to a Z-score of 1.96)
σ = population standard deviation
E = margin of error
Plugging in the given values, we have:
n = (1.96 * 8 / 1.5)²
Calculating this expression:
n = (15.68 / 1.5)²
n = 10.453³
n ≈ 111.32
Since you cannot have a fraction of a person, you should round up the minimum sample size to the nearest whole number. Therefore, the minimum sample size the researcher needs to use is 112 people.
To find the minimum sample size required, we need to use the formula for sample size calculation:
n = (Z^2 * s^2) / E^2
Where:
n = sample size
Z = Z-score for the desired confidence level (95% confidence level corresponds to a Z-score of 1.96)
s = population standard deviation
E = maximum tolerable error margin
Plugging in the values:
n = (1.96^2 * 8^2) / 1.5^2
n = (3.8416 * 64) / 2.25
n = 246.3744 / 2.25
n ≈ 109.5
Since we need a whole number for the sample size, we round up to the next highest integer:
n = 110
Therefore, the minimum sample size the researcher needs to use is 110.