Suppose that the starting salaries of female CFO have a mean of $56,000 and a standard deviation of $12,000. The starting salaries of male CFO have a mean of $50,000 and a standard deviation of $10,000. A random sample of 50 female CFO’s and a random sample of 40 male CFO’s are selected.
Find
Find the expected value of the sample mean difference.
Find the standard error of the sample mean difference.
What is the probability that the sample mean salary of male CFO will not exceed that of the female CFO’s?
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.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability
related to the Z score.
To find the expected value of the sample mean difference, you need to calculate the difference between the mean salaries of female CFOs and male CFOs, which is $56,000 - $50,000 = $6,000. The expected value of the sample mean difference is equal to the population mean difference, which is $6,000.
To find the standard error of the sample mean difference, you need to calculate the standard deviation of the sample mean difference. The formula for the standard error of the sample mean difference is:
Standard Error = sqrt((sd1^2/n1) + (sd2^2/n2))
where sd1 and sd2 are the standard deviations of the two populations (female CFOs and male CFOs) and n1 and n2 are the sample sizes.
In this case, the standard deviation of female CFOs is $12,000, the standard deviation of male CFOs is $10,000, the sample size of female CFOs is 50, and the sample size of male CFOs is 40.
Plugging these values into the formula, we get:
Standard Error = sqrt((12000^2/50) + (10000^2/40)) = $1699.33 (approximately)
Therefore, the standard error of the sample mean difference is approximately $1699.33.
To find the probability that the sample mean salary of male CFO will not exceed that of the female CFOs, we need to compare the two sample means and their variability.
First, we want to find the difference in sample means:
Mean difference = Sample mean of female CFOs - Sample mean of male CFOs
Next, we want to calculate the standard error of the sample mean difference using the formula mentioned earlier.
Finally, we can use a statistical test called the t-test to find the probability. We will compare the sample mean difference to the standard error of the sample mean difference using the t-distribution.
Specifically, we will calculate the t-value using the formula:
t-value = (Mean difference - 0) / Standard error of the sample mean difference
In this case, since we want to find the probability that the sample mean salary of male CFOs will not exceed that of female CFOs, we are looking for the probability of the t-value being less than zero.
We can look up this probability in a t-distribution table, or use statistical software to find the cumulative probability. The degrees of freedom for the t-distribution in this case would be the smaller of (n1 - 1) and (n2 - 1).
With the given sample sizes (n1 = 50, n2 = 40), and assuming the samples are independent and normally distributed, you can use the t-distribution to find the probability that the sample mean salary of male CFOs will not exceed that of the female CFOs.