Question

Juries should have the same racial distribution as the surronding communities. According to U.S. Census, about 18% of residence in Minneapolis, Minnesota are African American. Suppose that a local court randomly selects 100 adult citizens of Minneapolis to participate in the jury. If the sample proportion for African American is of interest,

a) can you apply the central limit theorem to find the sampling distribution for sample proportion( check conditions) ?

b) what is the mean of sampling distribution of sample proportion?

c) what is the standard error of sampling distribution of sample proportion?

d) by using empirical probability what is the probability that the proportion of available African Americans Jurors is more than two standard error from the population value of 0.18?

Answer 1

a) n = 100

    p = 0.18

np(1 - p) = 100 * 0.18 * (1 - 0.18) = 14.76

Since np(1 - p) > 10, so the sampling distribution of the sample proportion will be approximately normally distributed.

b) = p = 0.18

c) = sqrt(p(1 - p)/n)

           = sqrt(0.18 * (1 - 0.18)/100)

           = 0.0384

d) P(0.1032 < < 0.2568)

= P((0.1032 - )/ < ( - )/ < (0.2568 - )/)

= P((0.1032 - 0.18)/0.0384 < Z < (0.2568 - 0.18)/0.0384)

= P(-2 < Z < 2)

= P(Z < 2) - P(Z < -2)

= 0.9772 - 0.0228

= 0.9544