Number of Jurors by race:
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
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.
To determine the number of degrees of freedom for the asymptotic distribution of T_n, we need to calculate it based on the proportions of the different categories.
In this case, we have 4 categories: white, black, Hispanic, and other, with a total of 275 jurors.
Degrees of freedom (df) = Number of categories - 1
df = 4 - 1 = 3
So, the number of degrees of freedom for the asymptotic distribution of T_n is 3.
Next, we need to evaluate T_275, the test statistic, using the given data set.
T_275 = (Observed frequency - Expected frequency)² / Expected frequency
Expected frequency for each category = Total number of jurors * Proportion of category
Expected frequency for white category = 275 * 0.72 = 198
Expected frequency for black category = 275 * 0.07 = 19.25
Expected frequency for Hispanic category = 275 * 0.12 = 33
Expected frequency for other category = 275 * 0.09 = 24.75
T_275 = [(205-198)²/198] + [(26-19.25)²/19.25] + [(25-33)²/33] + [(19-24.75)²/24.75]
T_275 = [49/198] + [44.77/19.25] + [64.52/33] + [17.75/24.75]
T_275 = 0.247 + 2.321 + 1.952 + 0.717
T_275 = 5.237 (rounded to 3 decimal places)
Now, we need to find the p-value of this test using the chi-squared distribution. The p-value represents the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true (i.e., the jurors are representative of the population in the county).
To find the p-value, we can use statistical software or a chi-squared distribution table with 3 degrees of freedom. Given the importance of accuracy in the final answer, it is recommended to use reliable statistical software such as R.
The p-value for T_275 = 5.237 (rounded to 3 decimal places) can be calculated using software or a chi-squared distribution table with 3 degrees of freedom.
Please note that the p-value cannot be calculated accurately without accessing a chi-squared distribution table or using statistical software.