2 The article Statistical Evidence of Discrimination J Amer

2. The article Statistical Evidence of Discrimination (J Amer. Stat. Assoc.) discusses the case Swain v. Alabama (1965). in which it was alleged that there was discrimination against blacks in grand jury selection. Census data suggested that 25% of those eligible for jury duty were black, yet a random sample of 1050 called to appear for possible duty yielded only 177 blacks. a) Using an a = .01 test, does this data argue strongly for a conclusion of discrimination? b) Describe a Type 1 error for this question and the consequences of committing such an error. c) Do the same for a Type II error.

Solution

(a) The test hypothesis:

Ho: p=0.25 (i.e. null hypothesis)

Ha: p not equal to 0.25 (i.e. alternative hypothesis)

The test statistic is

Z=(phat-p)/sqrt(p*(1-p)/n)

=(177/1050-0.25)/sqrt(0.25*0.75/1050)

=-6.09

It is a two-tailed test.

Given a=0.01, the critical values are Z(0.005) =-2.58 or 2.58 (from standard normal table)

The rejection regions are if Z<-2.58 or Z>2.58, we reject the null hypothesis.

Since Z=-6.09 is less than -2.58, we reject the null hypothesis.

So we can conclude that this data argue strongly for a conclusion of discrimination

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(b)Type I error: We reject the null hypothesis when it is true.

We reject that 25% of those eligible for jury duty were black when it is true.

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(C) Type II error: We do not reject the null hypothesis when it is false.

We do not reject that 25% of those eligible for jury duty were black when it is false.