Question

4. An independent agency has been charged with investigating gender discrimination based on pay differentials in a large company. They find that from a sample of 236 women in salaried positions, women are paid an average of 48,352 dollars annually with a standard deviation of 5,285 dollars. From a sample of 257 men, they find that men in salaried positions on average earn about 54,285 dollars annually with a standard deviation of 8,456 dollars.

1. Calculate a 95 percent confidence interval for women.

2. What does this mean?

3. Calculate a 95 percent confidence interval for men.

4. What does this mean?

5. What can we conclude about gender discrimination in this company from the calculations above?

Answer 1

Given that from a sample of 236 women in salaried positions, women are paid an average of 48,352 dollars annually with a standard deviation of 5,285 dollars.

Since the sample is large greater than 30 hence assuming the normal distribution but the population standard deviation is not given hence t-distribution is applicable for confidence interval calculation.

a) The confidence interval is calculated as:

μ = M ± t(sM)

where:

M = sample mean

n = sample size

df = n-1, degree of freedom
t = t statistic determined by confidence level and the degree of freedom
sM = standard error = √(s2/n)

Calculation:

M = 48352
n = 236

df = 236-1= 235

t = 1.97 calculated using the excel formula for t-distribution which is =T.INV.2T(0.05, 235)
sM = √(52852/236) = 344.02

μ = M ± t(sM)
μ = 48352 ± 1.97*344.02
μ = 48352 ± 677.77

So, the confidence interval is:

{47674.23, 49029.77}

b) Interpreation:

Now based on the confidence interval calculation above with 95% confidence we can say that the population mean (μ) falls between 47674.23 and 49029.77.

c) Given that a sample of 257 men, they find that men in salaried positions on average earn about 54,285 dollars annually with a standard deviation of 8,456 dollars.

Since the sample is large greater than 30 hence assuming the normal distribution but the population standard deviation is not given hence t-distribution is applicable for confidence interval calculation.

The confidence interval is calculated as:

μ = M ± t(sM)

where:

M = sample mean

n = sample size

df = n-1, degree of freedom
t = t statistic determined by confidence level and the degree of freedom
sM = standard error = √(s2/n)

Calculation:

M = 54285
t = 1.97 calculated using the excel formula for t-distribution which is =T.INV.2T(0.05, 256)
sM = √(84562/257) = 527.47

μ = M ± t(sM)
μ = 54285 ± 1.97*527.47
μ = 54285 ± 1038.73

Thus the confidence interval is:

[53246.27, 55323.73].

d) Interpreation:

Now based on the confidence interval calculation above with 95% confidence we can say that the population mean (μ) falls between 53246.27 and 55323.73.

e) Based on the confidence interval of women that is {47674.23, 49029.77} and of men that is [53246.27, 55323.73] we can clearly make a conclusion that the Upper limit of the Women confidence interval is less than the Lower limit of the men confidence interval hence there is a clear indication of gender discrimination in this company from the calculations above.