Question

Among the dormitory rooms of a large, imaginary university, 9 percent tested positive for highly polluted air, 27 percent for moderately polluted air, and 41 percent for slightly polluted air. Of the 2,350 randomly sampled rooms where students reside, 250 had highly polluted air, 750 moderately polluted air, and 200 slightly polluted air. We have the following summary table:

Rooms Highly Polluted	Rooms Moderately Polluted	Rooms Slightly Polluted	Other (or Unpolluted) Rooms	Total
250	750	200	1150	2350

(i)

How can we express the hypotheses for testing whether students are proportionately (or disproportionately) assigned to these categories of residence hall rooms?

(ii)

What type of test would we use to investigate a question such as this?

(iii)

For this type of test, are the requisite assumptions and conditions satisfied?

(iv)

Carry out a hypothesis test. What is the value of your test statistic? Do the results yield solid evidence that students tend to be assigned disproportionately to halls with particular categories of pollution level?

Among the answer choices that follow, CHOOSE THE BEST COMBINATIONS OF THE OPTIONS GIVEN IN (i) AND (ii) AND (iii) AND (iv).

Group of answer choices

a.

(i)

H_0: Students are distributed among university residence halls rooms according to the distribution of the pollution categories comprised by the rooms.

H_A: The distribution of students among university residence halls rooms departs with statistical discernibility from the distribution of pollution categories comprised by the residence halls rooms.

(ii)

We can use a Chi-square goodness-of-fit test.

(iii)

Yes

(iv)

1320.357; the evidence is solid

b.

(i)

H_0: Students are distributed among the university’s residence halls rooms according to the distribution of the pollution categories comprised by the rooms.

H_A: The distribution of students among the university’s residence halls rooms is exactly as given by the sample distribution of rooms.

(ii)

We can use a Chi-square goodness-of-fit test.

(iii)

Yes

(iv)

1320.357; the evidence is solid

c.

(i)

H_0: Students are distributed among university residence halls rooms in accordance with a binomial distribution with mean proportion 0.489 and standard error 0.010312.

H_A: The distribution of students among university residence halls rooms departs with statistical discernibility from the binomial distribution of pollution categories comprised by the residence halls rooms.

(ii)

We can use a normal approximation to the binomial test.

(iii)

No

(iv)

12.84; the evidence is solid

d.

a.

(i)

H_0: Students are distributed among university residence halls rooms in accordance with a multinomial distribution where the mean count for each pollution category is the product of the population percentage of that pollution category and the size of the residence halls sample.

H_A: The distribution of students among university residence halls rooms departs with statistical discernibility from the multinomial distribution of pollution categories comprised by the residence halls rooms.

(ii)

We can use Pearson’s Chi-square test to investigate this question.

(iii)

No

(iv)

0.17; the evidence is weak

e.

Nothing in a. through d., above, is correct.

Answer 1

using minitab>stat>tables

we have

Chi-Square Goodness-of-Fit Test for Observed Counts in Variable: frequancy

Using category names in Rooms polluted

Test Contribution
Category Observed Proportion Expected to Chi-Sq
Highly 250 0.09 211.5 7.008
Moderately 750 0.27 634.5 21.025
Slightly 200 0.41 963.5 605.015
Other 1150 0.23 540.5 687.309

N DF Chi-Sq P-Value
2350 3 1320.357 0.000

option A is true

a.

(i) H_0: Students are distributed among university residence halls rooms according to the distribution of the pollution categories comprised by the rooms.

H_A: The distribution of students among university residence halls rooms departs with statistical discernibility from the distribution of pollution categories comprised by the residence halls rooms.

(ii) We can use a Chi-square goodness-of-fit test.

(iii) Yes

(iv) 1320.357; the evidence is solid