determine the size of the sample to be taken from a population with S.D. 10 if the probability of the sample mean differing from the population mean by more than 2, is less than 0.2
To determine the size of the sample to be taken from a population, given a desired probability of the sample mean differing from the population mean by more than a certain value, you need to use the formula for sample size calculation based on the standard deviation.
The formula for calculating the sample size, denoted as "n", is given by:
n = (Z * σ / E)²
Where:
- n is the sample size
- Z is the Z-score corresponding to the desired level of confidence
- σ (sigma) is the standard deviation of the population
- E is the maximum acceptable difference between the sample mean and the population mean
In this case, you are given the standard deviation (σ) of the population as 10 and the maximum acceptable difference (E) as 2. However, you need to determine the appropriate Z-score to ensure that the probability of the sample mean differing from the population mean by more than 2 is less than 0.2.
To find the Z-score, you can use a standard normal distribution table or a statistical software. Since the given probability is less than 0.2, we want to find the Z-score that corresponds to a cumulative probability of 1 - 0.2 = 0.8 (80%).
Using a standard normal distribution table, you can find that the Z-score corresponding to a cumulative probability of 0.8 is approximately 0.84.
Now, substituting the values into the formula, we have:
n = (0.84 * 10 / 2)²
n = 4.2²
n ≈ 17.64
Rounding up to the nearest whole number, the sample size needed is 18.
Therefore, you would need a sample size of 18 to ensure that the probability of the sample mean differing from the population mean by more than 2 is less than 0.2.