Monthly incomes of employees at a particular company have a mean of $5954. The distribution of sample means for samples of size 70 is normal with a mean of $5954 and a standard deviation of $259. Suppose you take a sample of size 70 employees from the company and find that their mean monthly income is $5747. How many standard deviations is the sample mean from the mean of the sampling distribution?
A. 0.8 standard deviations above the mean
B. 0.8 standard deviations below the mean
C. 7.3 standard deviations below the mean
D. 207 standard deviations below the mean
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
To find out how many standard deviations the sample mean is from the mean of the sampling distribution, we can use the formula for the z-score:
z = (sample mean - mean of the sampling distribution) / (standard deviation of the sampling distribution)
Given that the mean of the sampling distribution is $5954, the sample mean is $5747, and the standard deviation of the sampling distribution is $259, we can plug these values into the formula:
z = ($5747 - $5954) / $259
z = -$207 / $259
z = -0.8
Therefore, the sample mean is 0.8 standard deviations below the mean of the sampling distribution.
The correct answer is B. 0.8 standard deviations below the mean.
To determine how many standard deviations the sample mean is from the mean of the sampling distribution, you can use the formula for Z-score:
Z = (X - μ) / (σ / √n)
where:
- Z is the number of standard deviations from the mean
- X is the sample mean
- μ is the mean of the sampling distribution
- σ is the standard deviation of the sampling distribution
- n is the sample size
Given:
- X = $5747 (sample mean)
- μ = $5954 (mean of the sampling distribution)
- σ = $259 (standard deviation of the sampling distribution)
- n = 70 (sample size)
Substituting the values into the formula:
Z = ($5747 - $5954) / ($259 / √70)
Z = ($-207) / ($259 / √70)
To calculate this value, divide -$207 by the fraction ($259 / √70):
Z = -207 / (259 / √70)
Z ≈ -207 / (259 / 8.3666)
Z ≈ -207 / 30.9583
Z ≈ -6.6756
Therefore, the sample mean is approximately 6.6756 standard deviations below the mean of the sampling distribution.
The closest answer choice is:
C. 7.3 standard deviations below the mean.
(Please note that the exact answer is not provided as it requires more decimal places for accuracy.)