Number of Jurors by race:

Question

White - 205 (0.72 proportion in country)
Black - 26 (0.07 proportion in country)
Hispanic - 25 (0.12 proportion in country)
Other - 19 (0.09 proportion in country)
Total - 275

Proportion of Race in Country
White - 0.72 proportion in country
Black - 0.07 proportion in country
Hispanic - 0.12 proportion in country
Other - 0.09 proportion in country
TOtal - 1
Design a \chi ^2 test to see whether or not the jurors selected are representative of the population in the county. Denote by T_{n} the test statistic for this test.

What is the number of degrees of freedom of the asymptotic distribution of T_{n}. In other words,

\displaystyle \displaystyle T_ n\xrightarrow [n\to \infty ]{(d)}\chi ^2_{\ell }
for l=\quad
unanswered
Evaluate T_{275} on the given data set. (Answer accurate to 2 decimal places. )

T_{275}=\quad
unanswered
What is the p-value of this test?
(Answer accurate to 2 decimal places.)

You could use this tool or software such as R to find the quantiles of a chi-squared distribution.)

p-value

Answer 1

The degrees of freedom (df) of the asymptotic distribution of Tn can be calculated using the formula:

df = (number of categories - 1) = (4 - 1) = 3

Now, to evaluate T275, we need to calculate the expected frequencies under the assumption that the jurors selected are representative of the population in the county. We can do this by multiplying the proportion of each race in the country by the total number of jurors (275).

Expected Frequencies:
White: 0.72 x 275 = 198
Black: 0.07 x 275 = 19.25
Hispanic: 0.12 x 275 = 33
Other: 0.09 x 275 = 24.75

Next, we calculate the chi-square statistic using the formula:

Tn = Σ ((Observed frequency - Expected frequency)^2 / Expected frequency)

T275 = ((205 - 198)^2 / 198) + ((26 - 19.25)^2 / 19.25) + ((25 - 33)^2 / 33) + ((19 - 24.75)^2 / 24.75)

Finally, the p-value of this test can be found by comparing the chi-square statistic (T275) to the chi-square distribution with degrees of freedom (df) of 3.

Using a chi-square distribution table or software, we can find the p-value associated with T275.