7.7.STS.Handout.Key

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Material adapted from Skew The Script (skewthescript.org) *All solutions and teacher notes in blue* AP Statistics Handout Key : Lesson 7.7 Topics: one vs. two sample inference, two-sample z-interval for a difference of proportions Lesson 7.7 Guided Notes In a famous study, 1 investigators created mock identical resumés, which were sent to job placement ads in Chicago and Boston. Each resumé was randomly assigned either a commonly-white or commonly- black name. In total, 246 out of 2445 commonly-white named resumés received a callback and 164 out of 2445 commonly-black named resumés received a callback. One vs. Two Sample Inference Study results Commonly- White Names Commonly- Black Names Total Called back 246 164 410 Not called back 2199 2281 4480 Total 2445 2445 4890 1) Find the following quantities (show any calculations): 𝑛 1 , 𝑛 2 , 𝑝̂ 1 , 𝑝̂ 2 , 𝑝̂ 1 − 𝑝̂ 2 n 1 = 2445 n 2 = 2445 𝐩 ̂ ? = ??? ???? = ?. ??? 𝐩 ̂ ? = ??? ???? = ?. ??? 𝐩 ̂ ? − 𝐩 ̂ ? =. ??? 2) Are the proportions (who got called back) different enough to show convincing evidence of discrimination? Or do you believe this difference could be due to chance alone? Justify using your intuition. No calculations required. Answers will vary. Emphasize intuition about effect size (larger difference p ̂ 1 − p ̂ 2 means more likely to be significant) and sample size (larger n 1 & n 2 means more likely to be significant). 1 Bertrand, Marianne and Sendhil Mullainathan. "Are Emily And Greg More Employable Than Lakisha And Jamal? A Field Experiment On Labor Market Discrimination," American Economic Review , 2004, v94(4,Sep), 991-1013. https://www.nber.org/papers/w9873 Let s define: 𝑝̂ 1 = proportion of commonly-white name apps that got callback. 𝑝̂ 2 = proportion of commonly-black name apps that got callback.
2 Material adapted from Skew The Script (skewthescript.org) One vs Two Sample Situations One-sample situations: you compare a statistic in one population against a claim about that population. Two-sample situations: you measure the same statistic in two populations/treatments and see if they are significantly different. 3) Give an example of a one-sample situation and of a two-sample situation. One-sample situation: Someone claims that 5% or more of Duracel batteries are faulty. Is that true? Two-sample situation: Is the proportion of Duracel batteries that are faulty higher than the proportion of Energizer batteries that are faulty? Two-Sample Z-Interval for a Difference of Proportions Hypotheses: We don’t need to set up hypotheses to construct a confidence interval, but doing so is going to help us conceptualize why our interval is useful. 𝐻 0 : 𝑝 1 = 𝑝 2 𝐻 𝐴 : 𝑝 1 > 𝑝 2 Where: 𝑝 1 is the proportion of all applicants with commonly- white names who’d receive callbacks when applying to jobs like the ones in this study. 𝑝 2 is the proportion of all applicants with commonly- black names who’d receive callbacks when applying to jobs like the ones in this study. 4) Rewrite these hypotheses in a more mathematically convenient way: 𝐻 0 : 𝑝 1 − 𝑝 2 = 0 𝐻 𝐴 : 𝑝 1 − 𝑝 2 > 0 The alternative ( research ) hypothesis: there is discrimination, in which case the commonly-white named applications received a higher rate of callbacks. The null ( default/dull ) hypothesis: there is no discrimination , so the callback rate is the same in both groups. You’re seeing if there’s evidence to reject this claim.
3 Material adapted from Skew The Script (skewthescript.org) Making the Interval 5) Calculate and interpret the 95% confidence interval for the true difference in callback rates (white black), by following the steps below: b) Find the 95% interval. Sketch the interval on the top number line. Formula: 𝑝̂ 1 - 𝑝̂ 2 ± 1.96(𝜎) . Confidence Interval: .034 ± 1.96(0.0079) = (1.85%, 4.95%) c) Interpret your interval. Then, comment on whether your interval provides convincing evidence of a higher callback rate for commonly white names. Interpretation: We are 95% confident the interval from 1.85% to 4.95% captures the true difference in proportion of callbacks for resumés with commonly white vs. black names (among jobs similar to the ones in this study). Since 0 is outside our interval, it’s not plausible to assume 𝑝 1 = 𝑝 2 . Therefore, we have convincing evidence that commonly white name resumés receive a higher callback rate (among jobs similar to the ones in this study). 95% Interval C% Interval 95% Confidence Interval Formula 95% One-Sample Interval Formula 95% Two-Sample Interval Formula statistic ± margin of error 𝑝̂ ± ?. 𝟗? ∗ 𝑝̂(1 − 𝑝̂) 𝑛 (𝑝̂ 1 - 𝑝̂ 2 ) ± ?. 𝟗? ∗ √ 𝑝̂ 1 (1 − 𝑝̂ 1 ) 𝑛 1 + 𝑝̂ 2 (1 − 𝑝̂ 2 ) 𝑛 2 C% Confidence Interval Formula C% One-Sample Interval Formula C% Two-Sample Interval Formula statistic ± margin of error 𝑝̂ ± 𝒛 𝑝̂(1 − 𝑝̂) 𝑛 (𝑝̂ 1 - 𝑝̂ 2 ) ± 𝒛 ∗ √ 𝑝̂ 1 (1 − 𝑝̂ 1 ) 𝑛 1 + 𝑝̂ 2 (1 − 𝑝̂ 2 ) 𝑛 2 ?. 𝟗? : the critical value for a 95% z-interval. It means we’re include ~2 standard errors in the interval 𝒛 : the critical value of the z-interval. Tells you how many standard errors you’re including in your interval. a) The formula for the sampling distribution is given below. Find the parameters and sketch the sampling distribution curve on the bottom number line. ~ Norm (𝜇 = 𝑝̂ 1 - 𝑝̂ 2 , 𝜎 = √ 𝑝̂ 1 (1−𝑝̂ 1 ) 𝑛 1 + 𝑝̂ 2 (1−𝑝 ̂ 2 ) 𝑛 2 ) ~ Norm (𝜇 = .034, 𝜎 = √ .101 (1−.101) 2445 + .067 (1−.067) 2445 ) ~ Norm (𝜇 = .034, 𝜎 = 0.0079) Interval Formulas:
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4 Material adapted from Skew The Script (skewthescript.org) The Four Step Process for Inference ( State-Plan-Do-Conclude ) In the Bertrand-Mullainathan race/resumé study, mock identical resumés were sent to job placement ads in Chicago and Boston. Each resumé was randomly assigned either a commonly-white or commonly- black name. In total, 246 out of 2445 commonly-white named resumés received a callback and 164 out of 2445 commonly-black named resumés received a callback. a) Construct and interpret a 95% confidence interval for the difference in callback rates for commonly white vs. black named resumes (among jobs like the ones in this study). For “DO” phase: Calculator steps for two -sample z-interval for proportions: Conclude: We are 95% confident the interval from .018 to .049 captures the true difference in proportion of callbacks for resumés with commonly white vs. black names (among jobs similar to the ones in this study). b) Use your interval to draw a conclusion about whether callback rates differ by commonly white vs. black name. Since 0 is outside our interval, it’s not plausible to assume 𝑝 1 = 𝑝 2 . Therefore, we have convincing evidence that commonly-white name resumés receive a higher callback rate (among jobs similar to the ones in this study). State: We are constructing a 95% confidence interval for 𝑝 1 − 𝑝 2 . Where: 𝑝 1 is the proportion of all applicants with commonly- white names who’d receive callbacks when applying to jobs like the ones in this study. 𝑝 2 is the proportion of all applicants with commonly- black names who’d receive callbacks when applying to jobs like the ones in this study. Plan: We will calculate a two-sample z-interval for 𝑝 1 − 𝑝 2 , if all conditions are met. Random: Employers were randomly assigned either a commonly-white or commonly-black named resumé Large counts: (2445)(.101) ≥ 10 (2445)(.067) ≥ 10 246 ≥ 10 164 ≥ 10 2445(1 − .101) ≥ 10 2445(1 − .067) ≥ 10 2199 ≥ 10 2281 ≥ 10 Do: Two-prop z-interval: 𝑥 1 : 246, 𝑛 1 : 2445, 𝑥 2 : 164, 𝑛 2 : 2445, confidence level: 0.95 𝑝̂ 1 = 0.101 , 𝑝̂ 2 = 0.067 p ̂ 1 − p ̂ 2 = .034 Confidence interval: (0.018, 0.049) STAT TESTS B: 2- PropZInt… For groups 1&2: x: # of ‘successes’ n: sample size C-level: C% as decimal Calculate Output
5 Material adapted from Skew The Script (skewthescript.org) Lesson 7.7 Discussion Statistically Significant vs. Practically Important Statistically Significant Difference: Difference is too large to arise by chance alone . Confidence intervals and hypothesis tests tell us this Practically Important Difference: Difference is large enough to have real consequences in the world. Descriptive statistics and our experiences tell us this. Let’s look at the study results one more time: Commonly- White Names Commonly- Black Names Total Called back 246 164 410 Not called back 2199 2281 4480 Total 2445 2445 4890 Discussion Question: We found the difference between callback rates was statistically significant, but is it practically important? Look at the data, the callback rates, and the final confidence interval. Do you think the margin between black/white callback rates is practically important? Explain your reasoning. Many student s will use this line of reasoning: “It’s only a 3.4% difference – that doesn’t seem like a lot. So, it’s not practically important.” However, most statisticians would say this difference is practically important. Here’s why: 1. First, let’s take a look at the overall callback rate, regardless of racial category: 2. Given that it’s tough for anyone to get a callback, differences of a few percentage points are actually relatively large in terms of the relative probability of receiving a callback: Overall callback rate: 𝑝̂ = 410 4890 = 8.4% Overall callback rate is low. So, small percent difference between groups may make an important difference. Commonly-black name callback rate: 6.7% 10.1% − 6.7% 6.7% = 50.7% Commonly-white names were 50% more likely to get callbacks! That’s practically important. As the original study authors point out, “a white name yields as many more callbacks as an additional eight years of experience” (pg. 3). 𝑝̂ 1 = 10.1% 𝑝̂ 2 = 6.7% 𝑝̂ 1 − 𝑝̂ 2 = 3.4% Confidence interval for difference in callback rates: (1.8%, 4.9%) Recommended discussion norms: skewthescript.org/discussion-norms
6 Material adapted from Skew The Script (skewthescript.org) Lesson 7.7 Practice Teachers: We recommend providing additional practice exercises from your AP Stats textbook or from prior AP Stats exams. The following textbook sections and AP exam questions are aligned to this lesson: The Practice of Statistics (AP Edition) , 4th-6th editions: section 10.1 o 6 th edition update (CED-aligned): section 8.3 Stats: Modeling the World (AP Edition) , 4 th /5th editions: ch 21, 3rd edition: ch 22 Statistics: Learning from Data (AP Edition) , 2nd edition: section 11.1 Advanced High School Statistics , section 6.2 AP Exam Free Response Questions (FRQs) : 2006 Form B Q2, 2016 Q5 (part c) Handout Key by statistics student Greyson Zuniga
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