Question

In one case that made it all the way to the Supreme Court, a defense lawyer in Michigan challenged the process of selecting the jury pool in the trial of his accused client. Here are the facts:

• About 7.28% of the citizens in the court’s jurisdiction were black.

• The jury pool had between 60 and 100 members, only 3 of whom were black.

1. Is there enough evidence that conclude that the court is biased? Carry out an appropriate significance test with or construct a 95% confidence interval to make a conclusion.

Answer 1

Solution

Approach

Let p be the population proportion of ‘black’ in the jury pool.

Construct 95% confidence interval for p based on the given data of 3 out 60 and 3 out of 100

Conclude after comparing the given 7.28% or 0.0728 with the two confidence intervals.

Now, to work out the CI’s

100(1 - α) % Confidence Interval for p is: p_cap ± Z_α/2[sq.rt{p_cap(1 –p_cap)/n}] where

Z_α/2 is the upper (α/2)% point of N(0, 1), pcap = sample proportion, and n = sample size.

Given 95% confidence, α = 5% or 0.05 and hence Z_α/2 is the upper 2.5% point of N(0, 1) = 1.96 [from Standard Normal Tables]

Case 1: 3 out of 60

p_cap = 3/60 = 0.05

95% CI: 0.05 ± 1.96[sq.rt{0.05(0.95)/60}]

= (0, 0.105)

Case 2: 3 out of 100

p_cap = 3/100 = 0.03

95% CI: 0.03 ± 1.96[sq.rt{0.03(0.97)/100}]

= (0, 0.063)

Thus, in Case 1, the 95% confidence interval holds 0.0728 implying that there is no bias, but in Case 2 the 95% confidence interval does not hold 0.0728 implying that there is bias.

As such it is difficult to make a firm conclusion. Answer

[upto 3 out of 80, it is fair to conclude no bias, but above 80, there is indication of bias.]

Summary of Excel Calculations

Case 1

n	60
x	3
pcap	0.05
1-pcap	0.95
α	0.05
1 - α/2	0.975
Zα/2	1.959964
F	0.055147
LB	-0.00515
UB	0.105147

Case 2

n	100
x	3
pcap	0.03
1-pcap	0.97
α	0.05
1 - α/2	0.975
Zα/2	1.959964
F	0.033434
LB	-0.00343
UB	0.063434

DONE