Census data for a certain county show that 19% of the adult...

Name: Problem 28, Chapter 5: STATS Modeling The World
Uploaded: 2021-03-14T23:49:39-08:00
Duration: 3 min 51 s
Channel: Sheryl Ezze
Description: Census data for a certain county show that 19% of the adult residents are Hispanic. Suppose 72 people are called for jury duty and only 9 of them are Hispanic. Does this apparent under representation of Hispanics call into question the fairness of the jury selection system? Explain.

Census data for a certain county show that 19% of the adult residents are Hispanic. Suppose 72 people are called for jury duty and only 9 of them are Hispanic. Does this apparent under representation of Hispanics call into question the fairness of the jury selection system? Explain.

Key Concepts

Random Sampling and Fair Representation

Random sampling ensures that every member of the population has an equal chance of being selected, which is a cornerstone of fairness in practices like jury selection. Observed imbalances in small samples may not necessarily indicate bias in the selection process but can be a product of randomness, emphasizing the need for statistical tests before drawing conclusions about fairness.

Statistical Significance

This concept involves determining whether an observed difference (such as the discrepancy between the expected and actual number of Hispanic jurors) is sufficiently large that it is unlikely to have occurred by chance. A statistically insignificant result suggests that the difference is within the range of normal sampling variability, rather than indicative of a systemic problem.

Sampling Variability

This concept refers to the natural fluctuation in sample data that occurs when a random sample is taken from a population. Even when the true proportion in the population is known, a sample may differ from this proportion because of chance alone, meaning that occasional deviations from expected values are normal.

Binomial Distribution

The binomial distribution is relevant when dealing with a fixed number of independent trials, each with two possible outcomes (such as a person being Hispanic or not). In the context of jury selection, it provides a framework for calculating the probability of observing various counts of individuals from a subgroup in a random sample, given the known population proportion.

Transcript

00:01 So we would be assuming that the proportion of hispanics is equal to 0 .19 for jurors.

00:08 And alternately, it's underrepresented.

00:12 It's actually smaller than that.

00:15 And we know that the sample size was 72 people called to the jury, and we're getting a p hat of 9 out of 72.

00:28 And 9 out of 72 is 1 8th, which is 0 .1 .1 .2.

00:32 125.

00:33 So we want to know if we have evidence to say that this is consistent.

00:38 If the actual proportion is 0 .19 of hispanics in the jury pool, and we just happen to get a lower one down here.

00:48 We want to know what that p value is.

00:49 So how likely is it if the distribution really does have 19 % is this underrepresenting? so what's that likelihood of getting this value less than or equal to? the 0 .125 if the actual proportion is 19%.

01:11 And so, oops, let me erase that.

01:15 Let's find the z value and that z value is gonna be 0 .125 minus 0 .19 and then down here.

01:25 To get that test statistic, we're gonna use the 0 .19 and 1 minus 0 .19, which is gonna be 0 .81.

01:33 And then the sample size of 72.

01:36 Also, if we check n times p and n times 1 minus p, we will find that both of these are both greater than or equal to 10.

01:45 If we take 19 % of 72 and 19 %, well, 1 minus 19 % of that, both of them are greater than equal to 10.

01:54 So that was checking.

01:55 It doesn't say something about random sample.

01:57 I think that's what we're trying to really find...

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