The monthly earnings of teachers is normally distributed with a mean of $3,000 and the standard deviation of $250. We select a sample of 87 teachers. The sampling distribution of the sample mean has an expected value and standard deviation of:
a.) 3,000 and 26.8
b.) 3,000 and 1.69
c.) 3,000 and 250
d.) 3,000 and 2.87
e.) 3,000 and 321.6
To find the expected value and standard deviation of the sampling distribution of the sample mean, we use the following formulas:
Expected value of the sampling distribution of the sample mean = Mean of the population = $3,000
Standard deviation of the sampling distribution of the sample mean = Standard deviation of the population / Square root of the sample size
The standard deviation of the population is given as $250, and the sample size is 87.
Plugging in these values into the formula, we get:
Standard deviation of the sampling distribution of the sample mean = $250 / √(87)
Approximating the square root of 87 as approximately 9.33, we can calculate:
Standard deviation of the sampling distribution of the sample mean ≈ $250 / 9.33 ≈ $26.8
Therefore, the correct answer is a.) 3,000 and 26.8.