Census data for a certain county show that 19​% of the adult residents are Hispanic. Suppose 74 people are called for jury duty and only 10 of them are Hispanic. Does this apparent underrepresentation of Hispanics call into question the fairness of the jury selection​ system? Explain.

To determine whether the underrepresentation of Hispanics in a jury pool calls into question the fairness of the jury selection system, we can use statistical methods to assess if the observed difference from the expected proportion is likely due to chance.

First, we calculate the expected number of Hispanics if the selection were perfectly proportional to the population: Expected Hispanics = 19% of 74 = 0.19 * 74 ≈ 14.06

We would expect about 14 Hispanics in the jury pool based on the proportion of the adult population.

We can then perform a hypothesis test, such as a chi-square test for goodness-of-fit, to see if the observed count (10 Hispanics) significantly deviates from the expected count (about 14 Hispanics). However, given only the provided counts without additional data, we cannot conclusively perform the test here.

Another simpler approach to understand whether there is underrepresentation is to look at the standard error (SE) for a proportion and determine if the observed proportion is within a reasonable range of the population proportion.

The standard error of a proportion is calculated as: SE = sqrt[(p * (1 - p)) / n] where p is the population proportion, and n is the sample size.

For this scenario: SE = sqrt[(0.19 * (1 - 0.19)) / 74] ≈ sqrt[0.1539 / 74] ≈ 0.045

Typically, we look at a 95% confidence interval, which would be the expected proportion plus or minus 1.96 times the standard error.

Lower bound = 0.19 - (1.96 * 0.045) ≈ 0.102 Upper bound = 0.19 + (1.96 * 0.045) ≈ 0.278

This interval shows the range of proportions we would expect to see 95% of the time from random sampling.

The observed proportion of Hispanics in the jury pool is 10 out of 74, or approximately 0.135.

Since 0.135 is within the calculated bounds of 0.102 and 0.278, the observed proportion is within what we might expect due to chance variation, indicating that the underrepresentation of Hispanics in this particular jury may not be sufficient evidence on its own to question the fairness of the jury selection system. However, this is a simplified analysis and further statistical tests could be conducted for more conclusive results.