Sex Discrimination - The Data (1/2)
The sex discrimination study can be thought of as 48 red and white cards.
| sex | promoted | not promoted | Total |
|---|---|---|---|
| male | 21 | 3 | 24 |
| female | 14 | 10 | 24 |
| Total | 35 | 13 | 48 |
Alex John Quijano
09/27/2021
In the past lectures, we learned about the following:
Population: Refers to all individuals or objects for particular interest of study. A value referring to the population is called a parameter.
Sample: Refers to a part (or subset) of a population.A value referring to the sample is called a statistic.
Observational Study: Outcomes are measured without introducing a control to affect the outcome.
Experimental Study: Outcomes are measured while introducing a control to study an unknown outcome.
Simple Random Sampling: Randomly sampling - with replacement or without replacement - from a population (or existing data).
The basics of the the central limit theorem - the means of sufficiently large random samples from the population with replacement approaches a normal distribution, a type of probability distribution with a shape of a bell curve.
In this lecture, we will learn about the following:
Formulating/Identifying the null hypothesis.
Formulating/Identifying the alternative hypothesis (claim).
Formulating/Identifying the parameter and statistic in a given research question.
The idea of randomization test to produce a probability distribution under the null hypothesis.
Sources:
This example is from your textbook, OpenIntro: IMS Section 11.1.
The example in this lesson is taken from a paper on the Rosen, Benson, and Thomas H Jerdee. 1974. “Influence of Sex Role Stereotypes on Personnel Decisions.” Journal of Applied Psychology 59 (1): 9..
Participants: 48 male bank supervisors at the University of North Carolina in 1972.
Task: Assume the role of the personnel director of a bank and were given a personnel file to judge whether the person should be promoted to a branch manager position.
What are the files? half of them are “male” and the other half are “female”, all identical files.
These files were randomly assigned to the subjects.
For each supervisor both the sex associated with the assigned file and the promotion decision were recorded.
The sex discrimination study can be thought of as 48 red and white cards.
|
decision
|
|||
|---|---|---|---|
| sex | promoted | not promoted | Total |
| male | 21 | 3 | 24 |
| female | 14 | 10 | 24 |
| Total | 35 | 13 | 48 |
The sex discrimination study can be thought of as 48 red and white cards.
Observed Difference in proportion is 87.5% (male) \(-\) 58.3% (female) \(=\) 29.2%.
The large difference in promotion rates suggest there might be discrimination against women in promotion decisions.
sex and decision.The difference in promotion rates of 87.5% \(-\) 58.3% \(=\) 29.2%. This observed difference is what we call a point estimate of the true difference.
sex and decision are independent. They have no relationship, and the observed difference between the proportion of males and females who were promoted, 29.2%, was due to the natural variability inherent in the population.sex and decision are not independent. The difference in promotion rates of 29.2% was not due to natural variability, and equally qualified female personnel are less likely to be promoted than male personnel.A point estimate is a sample statistic if the obtained parameter is a single value.
The difference in promotion rates of 87.5% \(-\) 58.3% \(=\) 29.2%. This observed difference is what we call a point estimate of the true difference.
These hypotheses are part of what is called a hypothesis test, a statistical technique used to evaluate competing claims using data.
This hypothesis assumes that any differences seen are due to the variability inherent in the population and could have occurred by random chance.
The data shows that 35 bank supervisors recommended promotion and 13 did not.
Now, suppose the bankers’ decisions were independent of the sex of the candidate.
Simulating a world with no sex discrimination, which means that the mean difference between the proportion of males and females who were promoted.
We can actually perform this randomization, which simulates what would have happened if the bankers’ decisions had been independent of sex but we had distributed the file sexes differently.
In the simulation, we shuffle the 48 personnel files, 35 labelled promoted and 13 labelled not promoted, together and we deal files into two new stacks.
Assumption under the null hypothesis: 35 of the bank managers would have promoted the individual whose content is contained in the file independent of the sex indicated on their file.
The sex discrimination data is shuffled and reallocated to new groups of male and female files.
We summarize the randomized data to produce one estimate of the difference in proportions given no sex discrimination. Note that the sort step is only used to make it easier to visually calculate the simulated sample proportions.
This plot shows the distribution of the differences in proportions
\(\frac{2}{100}\) of the shuffles (shown in blue dots) has difference in proportion larger than our observed statistic.
Notice that the distribution exhibits the normal distribution (the central limit theorem).
Regarding the distribution of the differences in proportions:
Note that the distribution of these simulated differences in proportions is centered around 0 (null statistic).
Under the null hypothesis our simulations made no distinction between male and female personnel files.
The difference of 29.2% (observed statistic) is a rare event if there really is no impact from listing sex in the candidates’ files, which provides us with two possible interpretations of the study results:
If \(H_0,\) the Null hypothesis is true: Sex has no effect on promotion decision, and we observed a difference that is so large that it would only happen rarely.
If \(H_A,\) the Alternative hypothesis is true: Sex has an effect on promotion decision, and what we observed was actually due to equally qualified female candidates being discriminated against in promotion decisions, which explains the large difference of 29.2%.
Statistical inference is the practice of making decisions and conclusions from data in the context of uncertainty.
Note: Sometimes uncertainty can not be quantified precisely but it can be quantified. Uncertainty is the estimation of error present in data and it can give us some level of confidence on how sure we are about our estimation about the population.In this lecture we talked about:
Formulating/Identifying the null hypothesis and the alternative hypothesis.
Formulating/Identifying the parameter and statistic in a given research question.
The idea of randomization test to produce a probability distribution under the null hypothesis.
In the next lecture,
More examples on null and alternative hypothesis (means, proportions, etc).
An introduction to the p-value.
Within your group, discuss the answers for the following problem.
Hypotheses. For each of the research statements below, note whether it represents a null hypothesis or an alternative hypothesis. This exercise is from your textbook OpenIntro: IMS Section 11.5
The number of hours that grade-school children spend doing homework predicts their future success on standardized tests.
King cheetahs on average run the same speed as standard spotted cheetahs.
For a particular student, the probability of correctly answering a 5-option multiple choice test is larger than 0.2 (i.e., better than guessing).
The mean length of African elephant tusks has changed over the last 100 years.
The risk of facial clefts is equal for babies born to mothers who take folic acid supplements compared with those from mothers who do not.
Caffeine intake during pregnancy affects mean birth weight.
The probability of getting in a car accident is the same if using a cell phone than if not using a cell phone.