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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: