please see the attached file.

The company you work for employs 585 assembly-line workers - 526 men and 59 women. 16 men and 10 women have been laid off.

Of the 526 male assembly-line workers, 134 were over 40 years of age. 12 of the 16 laid off men are over 40 years of age. Those 12 men are alleging that they were discriminated against because of their age.

a) Construct the strongest possible argument that the 12 laid off men over 40 years of age could make in favor of their assertion that they have been discriminated against.

They can argue that they were discriminated against if they were disproportionately laid off from the company. Of the male assembly line workers, 25.5% are over 40 (134/526 = 0.25475), and of the men who were laid off, 75% are over 40 (12/16 = 0.75). It seems like there was bias against the older men because 75% is so much larger than 25.5%, but we have to do a statistical test to make sure.

We're comparing proportions, so we have to use a z-test. It is a two-proportion z-test because we're comparing two proportions instead of one proportion to a fixed number.

In this case, the z-statistic is calculated like this:

The standard deviation is calculated as:

So, let's plug in the numbers from this example:

p-hat = (134 + 12)/(526 + 16) = 146/542 = 0.26937
q-hat = 1 - 0.26937 = 0.73063

sp = √(0.26937)(0.73063)(1/526)(1/16) = 0.00484

z = (0.25475 - 0.75)/0.00484 = -102.324

We want to do a one-tailed test (test that one proportion is larger than the other), and this z-score is highly significant. The p-value is less than 0.001.

Therefore, you can reject the null hypothesis that the proportion of men over 40 who were laid off is less than or equal to the proportion of men over 40 in the company, and you can accept the null hypothesis that men over 40 were disproportionately laid off. There is evidence of discrimination.

b) The 134 male workers over 40 years of age earned a mean of $38,762, while the 392 male workers under the age of 40 earned a mean of $36,894. Does this information strengthen or weaken the argument made in a)? Give reasons for your answer.

This information weakens the argument made in a because the company might have laid off those workers, not because they were over 40, but because the company needed to cut costs and therefore laid off the people earning the highest salaries. Because people with higher salaries also tended to be older, then most of the laid off workers happened to be over 40.

It would be important to check if the difference in salaries was statistically significant, because if it wasn't, that might be evidence that they were fired based on age, not salary. However, we weren't given the standard deviations of the salaries, so we can't test for a difference in salary in this problem.

c) The company argues that the men could always have found a cut-off age above and below which the proportions of workers laid off would be different from one another. If it hadn't been 40, it might have been 35 or 30, or 45 or 50; but there would always be some age that would divide the workers into two groups that would appear to have been treated differently. Is this argument valid? Explain your answer.

This is not true. If people were laid off at random, then, for any age group, the proportion of people of that age should be the same in the laid off group and the company as a whole. So, for people over 40, since they make up 25% of the company, they should be expected to make up 25% of the people who were laid off. Of course, the proportions probably wouldn't be exactly the same, but they should be close enough that a statistical test like what we did in part a would not find the proportions significantly different.

If two age groups make up different percentages of the company's populations, then that will be reflected in the workers who are laid off, even if the lay offs are at random. For example, if workers in their 30's make up 30% of the population and workers in their 40's make up 15% of the population, then of the people who are laid off, twice as many people will be in their 30's as will be in their 40's. This is not a sign of bias against the younger workers, it just reflects their numbers in the company.