Statistical Inference Estimation Confidence Intervals Estimate the proportion of the electorate who support Candidate X Hypothesis Tests Make a decision Will Candidate X win the election?

Hypothesis Test: An example Castanada v. Partida 430 U.S. 482 Background Rodrigo Partida arrested and convicted of felony crime in Hidalgo County, Texas in 1972 Partida, Mexican-American, argued discrimination in selection of grand jury that indicted him Argued before the Supreme Court on Nov. 9, 1976 Filed habeas corpus petition alleging denial of due process and equal protection under 14th Amendment because of gross under-representation of Mexican-Americans on the county grand juries

Castanada v. Partida Statistical evidence presented to court Population Hidalgo County: 181,535 Mexican-American: 143,611 (79.1%) Grand Juries ( ) 870 persons summoned Mexican-American: 339 (39.0%)

Castanada v. Partida The population is all potential grand juries The sample is the jury data Let p be probability that a random juror is Mexican-American If no discrimination, then p =.791 (State’s position) Otherwise, p <.791 (Defendant’s position) Set up null and alternative hypotheses Ho: p =.791 vs. Ha: p <.791

Castanada v. Partida The hypotheses involve the population parameter p The evidence is based on the sample proportion The central question: If the state’s case is true and p =.791, what’s the chance of observing a sample proportion as small as.390? Central logic of hypothesis tests: Assume the null hypothesis is true and ask what’s the probability of observing what we actually did observe. That probability is called the P-value

Castanada v. Partida Computing the P-value (you won’t have to do it!) = -.401/.0138 = The z-score value is over 29 standard deviations from the mean! That’s a number with a decimal point and then 186 zeros before it gets to 6. It’s about the same chance of winning the Powerball Lottery 23 times in a row!

Castanada v. Partida Conclusion: reject the null hypothesis If there was no discrimination, and if the true probability of picking a Mexican-American juror was in fact 79.1%, then the chance of getting a jury composed of only 39% Mexican-American is astronomically miniscule. Conclusion: We reject chance as the explanation

Castanada v. Partida Legal Issue: finding of statistical significance does not constitute proof of discrimination But it’s used to establish prima facie case Burden of proof shifts to plaintiff to give another reasonable explanation for disparities on jury On March 23, 1977, Supreme Court ruled 5-4 in favor of Partida to establish his prima facie case.

Castanada v. Partida: Legal Issue Justice Blackmun: “As a general rule for such large samples, if the difference between the expected value and the observed number [of Mexican-Americans] is more than 2 or 3 standard deviations, then the hypothesis that the jury selection was random would be suspect by a social scientist.” Federal courts have fashioned “two or three standard deviation” norm into rule of law, particularly in discrimination cases

Summary Confidence interval estimates an unknown parameter Hypothesis test assesses the evidence for some claim about the value of an unknown parameter. Central Question: “Could the effect we see in the data just be an accident due to chance, or is it good evidence that the effect is really there in the population?” Answer by computing the probability (P-value) that the observed effect is as large as what we would expect by chance, assuming the null hypothesis were true. A small P-value means that the outcome is unlikely to occur by chance. This is evidence against the null hypothesis. A moderate or large P-value means the outcome could be consistent with chance as an explanation. There is insufficient evidence to reject the null hypothesis.