Suppose that, on average, electricians earn approximately $\mu=$54,200 per year in the United States. Assume that the distribution for electricians' yearly earnings is normally distributed and that the standard deviation is $\sigma=$11,500.
Given a sample of twenty-five electricians, what is the standard deviation for the sample mean?
a. 20,700
b. 2,300
c. 13,800
d. 4,600
Answer (1)
The standard deviation of the sample mean is calculated using the formula: σ x ˉ = n σ .
Substitute the given values: σ = 11500 and n = 25 .
Calculate the standard deviation of the sample mean: σ x ˉ = 25 11500 = 2300 .
The standard deviation for the sample mean is 2300 .
Explanation
Understand the problem and provided data We are given that the average yearly earning of electricians is μ = $54 , 200 , t h es t an d a r dd e v ia t i o n o f t h eye a r l ye a r nin g i s \sigma = $11,500, the earnings are normally distributed, and the sample size is n = 25 . We want to find the standard deviation of the sample mean.
State the formula for the standard deviation of the sample mean The standard deviation of the sample mean is given by the formula: σ x ˉ = n σ , where σ is the population standard deviation and n is the sample size.
Substitute the values and calculate Substitute the given values into the formula: σ x ˉ = 25 11500 = 5 11500 = 2300
State the final answer The standard deviation for the sample mean is $2 , 300 . Therefore, the correct answer is b.
Examples
Understanding the standard deviation of a sample mean is crucial in various real-world scenarios. For instance, if a researcher wants to estimate the average income of electricians in a specific region, they would collect a sample of electricians' incomes. The standard deviation of the sample mean helps the researcher understand the precision of their estimate. A smaller standard deviation indicates a more precise estimate, allowing for better informed decisions and policies related to wages and employment.
