Suppose that, on average, electricians earn approximately [tex]$\mu=$54,000[/tex] per year in the United States. Assume that the distribution for electricians' yearly earnings is normally distributed and that the standard deviation is [tex]$\sigma=$12,000[/tex].
Given a sample of four electricians, what is the standard deviation for the sample mean?
a. 36,000
b. 6,000
c. 54,000
d. 12,000
Answer (2)
The standard deviation of the sample mean is calculated using the formula: σ x ˉ = n σ .
Substitute the given values: σ = 12000 and n = 4 .
Calculate the standard deviation of the sample mean: σ x ˉ = 4 12000 = 2 12000 = 6000 .
The standard deviation for the sample mean is 6 , 000 .
Explanation
Understand the problem and provided data We are given that the average yearly earning of electricians is μ = $54 , 000 . The standard deviation of the yearly earning is σ = $12 , 000 . The distribution of the yearly earnings is normal, and we have a sample of n = 4 electricians. 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, also known as the standard error, 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 given values into the formula Substitute the given values into the formula: σ x ˉ = 4 12000 .
Calculate the standard deviation of the sample mean Calculate the value of σ x ˉ : σ x ˉ = 2 12000 = 6000 .
State the final answer Therefore, the standard deviation for the sample mean is $6 , 000 .
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, meaning the sample mean is likely closer to the true population mean. This concept is also vital in quality control, where manufacturers use sample means to ensure their products meet certain standards.
The standard deviation for the sample mean of the electricians' earnings is $6,000, calculated using the formula σ x ˉ = n σ . We substituted the population standard deviation of $12,000 and the sample size of 4 into the formula, resulting in a sample standard deviation of $6,000. Therefore, the correct answer is $6,000.
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