10D Preview

Preparing for the next class

In the next in-class activity, you will need to be able to identify the sample and  population, calculate sample proportions and their differences, and identify independent  vs. dependent samples. You will also need to know how to use technology to calculate  the confidence interval for the difference in two proportions from independent samples  and understand how to interpret the confidence interval in the context of the data.

Do job callbacks differ based on the perceived gender of the applicant? To examine this  question, we will analyze data from an experiment[1] to assess the impact of gender and  race on a job applicant receiving a callback, the opportunity to progress to the next part  of the applicant process based on their application. The researchers submitted  applications to job openings posted in Boston and Chicago in 2001 and 2002. The  researchers randomly assigned names that are commonly associated with particular  races and genders. The data are from the resume[2] dataset in the OpenIntro R  package.

Question 1

In this preview assignment, we will focus on whether there was a difference between  the proportion of callbacks for applications the researchers identified as being  perceived as female and the proportion of callbacks for applications the researchers  identified as being perceived as male.

Of the 3,746 applications with names perceived as female, 309 received callbacks.  Of the 1,124 applications with names perceived as male, 83 received callbacks.

  1. What is the population of interest?
    1. Adults who submit job applications
    2. Female adults who submit job applications
    3. 3,746 applicants in the data who were perceived to be female
    4. 4,870 applicants in the data
  2. What is the sample?
    1. Adults who submit job applications
    2. Female adults who submit job applications
    3. 3,746 applicants in the data who were perceived to be female
    4. 4,870 applicants in the data
  3. Calculate [latex]\hat{p}[/latex], the proportion of applicants in the data who received callbacks.
  4. Calculate [latex]\hat{p_{F}}[/latex], the proportion of applicants perceived as female who received  callbacks.
  5. Calculate [latex]\hat{p_{M}}[/latex], the proportion of applicants perceived as male who received  callbacks.

Question 2

We are interested in understanding [latex]p_{F} - p_{M}[/latex], the true difference between the  proportion of applicants perceived as female who received callbacks and the  proportion of applicants perceived as male who received callbacks.

  1. Calculate the associated sample statistic [latex]\hat{p_{F}} - \hat{p_{M}}[/latex], the best guess for the difference in proportions given our data.
  2. Based on your answer to Part a, do you think the proportions of callbacks  differ based on perceived gender? Explain.

Though we have an estimate, or a “best guess,” for the difference in proportions of applicants who received callbacks between the two groups, we expect there is some variability associated with that estimate. In other words, if we calculated the difference in the proportions of applicants who received callbacks from two other random samples of 3,746 applicants who were perceived as female and 1,124 applicants who were perceived as male, we would expect to get a different (yet probably close) value of [latex]\hat{p_{F}} - \hat{p_{M}}[/latex] compared to what we did in the previous exercise.

Similar to a single proportion, we can calculate a confidence interval to obtain a plausible range of values the true difference in proportions takes, assuming certain conditions hold. For the remainder of this assignment, you will focus on the independence condition and on using technology to calculate the confidence interval. In the in-class activity, you will learn more about the underlying mathematics and other conditions.

Question 3

There are two different methods for calculating confidence intervals for the  difference in proportions. The method you use depends on whether the two groups  are independent or dependent (paired). If the two groups are independent, the  sample for one group is drawn independently of the other group. Knowing the  observations of one group does not provide useful information about the other  sample. Additionally, the groups can be different sizes.

If the two groups are dependent (also known as paired), the samples for the two  groups are not drawn independently of one another. Knowing the observations of one group does provide useful information about the other sample. Additionally, both  groups must be the same size.

Below are a few analysis objectives. Identify which of the following involves samples  from independent groups. Select all that apply.

  1. Assess student learning by looking at the difference in the mean score of a  statistics test taken at the beginning and end of the semester
  2. Assess the difference in proportions of students who major in statistics between  public and private colleges
  3. Assess the difference in mean daily hours users spend on their phones between  iPhone and Samsung users
  4. Assess the difference in participants’ fitness levels before and after a six-week  training program
  5. Assess the difference in proportions of participants who pass a fitness test  between those who do a six-week training program and those who don’t

Question 4

Let’s go back to the job applicants analysis. What are the two groups in this study?  Are they independent or dependent? Select the best response.

  1. The two groups are those who received callbacks and those who didn’t. They are  independent, since the sample for one group was drawn independently of the  sample for the other group.
  2. The two groups are those who received callbacks and those who didn’t. They are  not independent, since the sample for one group gives useful information about  the sample for the other group.
  3. The two groups are those who were perceived as female and those who were  perceived as male. They are independent, since the sample for one group was  drawn independently of the sample for the other group.
  4. The two groups are those who were perceived as female and those who were  perceived as male. They are not independent, since the sample for one group  gives useful information about the sample for the other group.

Question 5

We will focus on calculating the confidence interval for the difference between two population proportions for independent groups. To calculate the confidence interval, use the Confidence Interval & Significance Test tab on the DCMP Compare Two Population Proportions tool: https://dcmathpathways.shinyapps.io/2sample_prop/.

To enter the summary data at the beginning of this preview assignment, select “Number of Successes” under “Enter Data” on the left-hand side. Since we are interested in calculating the confidence interval for [latex]\hat{p_{F}} - \hat{p_{M}}[/latex], Group 1 is the group perceived as female and Group 2 is the group perceived as male.

Enter the number of successes and sample sizes for each group in the tool. You can also add informative group labels. The following is a screenshot to help you get started.

Once you have entered the information for both groups, the confidence interval will  be calculated in the tool. You can change the confidence interval using the slider on the left-hand side.

What is the 95% confidence interval for the difference in proportions of applicants  who received callbacks between those perceived as female and those perceived as male?

Question 6

What does the 95% confidence interval mean? Select all that apply.

  1. We are 95% confident that the true difference in proportions of applicants who  received callbacks between those perceived as female and those perceived as  male is between −0.009 and 0.0263.
  2. We are 95% confident that the difference in proportions of applicants who  received callbacks between those perceived as female and those perceived as  male in the sample is between −0.009 and 0.0263.
  3. The confidence interval does not provide sufficient evidence of a difference in the  proportions of callbacks between the two groups since 0 is in the interval.
  4. The confidence interval does provide sufficient evidence of a difference in the  proportions of callbacks between the two groups since 0 is in the interval.

e) There is a 95% probability that the true difference in proportions of applicants who  received callbacks between those perceived as female and those perceived as  male is between −0.009 and 0.0263.

 

  1. Bertrand, M. & Mullainathan, S. (2004). Are Emily and Greg more employable than Lakisha and Jamal? A field experiment on labor market discrimination. American Economic Review 94(4), 991–1013. DOI: 10.1257/0002828042002561. http://www.nber.org/papers/w9873 ↵
  2. Which resume attributes drive job callbacks? (Race and gender under study.) (n.d.). OpenIntro. Retrieved from https://www.openintro.org/data/index.php?data=resume ↵