Question

1. A hypothetical investigation on rider satisfaction with a particular public transit system serving commuting residents...

1.

A hypothetical investigation on rider satisfaction with a particular public transit system serving commuting residents of British California (BC) and Prince Edward’s County (PEC) offers some interesting findings. The proportion of commuters from BC that indicated low satisfaction with the transit system’s service in the 2018 calendar year was 65 percent, and the proportion from PEC was 70 percent. These point estimates were based on samples of 5,380 BC commuters and 6,810 PEC commuters, whose system-using commuters number in the millions.

(i)
In order to compute a valid confidence interval, what values/conditions must be satisfied? (ii)

Assume that we may validly construct confidence intervals, and proceed with computing a 95% confidence interval for the difference between the proportions of British Californian commuters and Prince Edward County commuters who report low satisfaction with the public transit system.

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

a. (i)

The product of 5,380 and 65 percent must equal or surpass 10, as must the product of 5,380 and 0.35; samples from BC and PEC must be selected independently; the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes; and samples selection must be randomized.

(ii)
(-0.067, -0.033)

b. (i)

The product of 6,810 and 70 percent must equal or surpass 10, as must the product of 6,810 and 0.30; samples from BC and PEC must be selected independently; the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes; and samples selection must be randomized.

(ii)
(0.033, 0.067)

c.

(i)

The product of 5,380 and 65 percent must not surpass 10, true also for the product of 5,380 and 0.35; samples from BC and PEC must be selected independently; the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes; and samples selection must be randomized.

(ii)
(-0.066, -0.032)

d. (i)

The product of 6,810 and 70 percent must equal or surpass 10, as must the product of 6,810 and 0.30; samples from BC and PEC must be selected independently; the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes; and sampling must proceed according to a multi-stage cluster design.

(ii)
(0.033, 0.067)

e.
(i)
Each of what we see for (i) in a. and b. is correct.
(ii)
For (ii), the value given under a. is correct, but the value in b. is incorrect.

2.

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).

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.

3.

Consider again the situation described in Problem 2, above. There you were asked to compute a confidence interval.

Here, conduct a hypothesis test to ascertain whether the data given in Problem 2 offer strong evidence that the proportion of commuters from British California expressing low satisfaction with the transit system differs from the Prince Edward County proportion.

To do this problem correctly, make sure you consult Section 6.2.3, pages 219-221, in OpenIntro Statistics, 4th Edition.

(i)

In order to conduct the hypothesis test, what values/conditions must be satisfied?

(ii)

How do we express the null and alternative hypotheses?

(iii)

Go ahead with the test. What is the value on your test statistic?

(iv)

What is the conclusion of your test?

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

a. (i)

The product of 5,380 and 68 percent must equal or surpass 10, as must the product of 5,380 and 0.32; samples from BC and PEC must be selected independently; the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes; and samples selection must be randomized.

(ii)
H_0: p_BC = p_PEC H_A: p_BC < p_PEC

(iii)

-5.68

(iv)

We conclude that the proportion of commuters from BC giving low satisfaction ratings to the transit system is less than the proportion of PEC commuters giving low satisfaction ratings.

We conclude that the proportion of commuters from BC giving low satisfaction ratings to the transit system differs from the proportion of PEC commuters giving low satisfaction ratings.

c. (i)

The product of 6,810 and 68 percent must equal or surpass 10, as must the product of 6,810 and 0.32; samples from BC and PEC must be selected independently; the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes; and samples selection must be randomized.

(ii)
H_0: p_BC = p_PEC H_A: p_BC ≠ p_PEC (iii)
-5.78
(iv)

We conclude that the proportion of commuters from BC giving low satisfaction ratings to the transit system differs from the proportion of PEC commuters giving low satisfaction ratings.

d. (i)

The product of 6,810 and 68 percent must equal or surpass 10, as must the product of 6,810 and 0.32; samples from BC and PEC must be selected independently; the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes; and samples selection must be randomized.

(ii)

H_0: p_BC = p_PEC

H_A: p_BC < p_PEC

(iii)

-5.68

(iv)

We conclude that the proportion of commuters from BC giving low satisfaction ratings to the transit system is less than the proportion of PEC commuters giving low satisfaction ratings.

e. None of the above answers sets is correct.

A study compared taste ratings from three different methods for cooking old-fashioned, hot oatmeal. The three methods were 1) get the water boiling before you add the oatmeal, then add the oatmeal, and allow light boiling for 5 minutes; 2) soak the oatmeal overnight, and then begin boiling, and allow light boiling for 5 minutes; 3) simply add the oatmeal, add the water, and fire up the burner to achieve a boil, and allow light boiling for 5 minutes.

The study assigned at random 25 breakfasters to each method. After consuming the oatmeal, each sampled person gave a taste rating to the breakfast consumed.

(i)

What are the hypotheses for evaluating whether the average taste ratings are different for the different cooking methods?

(ii)

What are the degrees of freedom associated with the F-test for evaluating these hypotheses?

(iii)

Suppose we test at the level alpha = 0.05, and we find that the p-value for this test is 0.0135. What is the conclusion of the test?

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

a.
(i)
H_0: Method 1 Taste Rating = Method 2 Taste Rating = Method 3 Taste Rating

H_A: Each one of the methods yields a different taste rating. (ii)
dfG = 2, dfE = 72
(iii)

We conclude that there is sufficient evidence that each one of the methods associates with a different taste rating.

We conclude that there is sufficient evidence that at least one of the methods associates with a different taste rating.

c.

(i)

H_0: Method 1 Taste Rating = Method 2 Taste Rating = Method 3 Taste Rating

H_A: At least one of the methods yields a taste rating different from the other two.

(ii)

dfG = 3, dfE = 72

(iii)

We conclude that there is sufficient evidence that at least one of the methods associates with a different taste rating.

d.
(i)
H_0: Method 1 Taste Rating ≠ Method 2 Taste Rating ≠ Method 3 Taste Rating H_A: At least two of the methods yields the same taste rating.
(ii)
dfG = 2, dfE = 72
(iii)

We conclude that there is sufficient evidence that at least two of the methods associate with the same taste rating.

e) None of the above answers sets is correct.

4.

Consider a hypothetical experiment in which investigators randomly selected 58 men from the membership list of the Sacramento Valley Motorist Society and then randomly assigned each man either to a sports utility vehicle (each one the same model and year and with the same amount of cumulative miles) or to a sedan (each one also exactly like the other). The men, 29 in each group, were instructed to drive these cars 100 miles on Interstate 5 in California, ostensibly to test for comfort of the driving experience. However the investigators’ real interest was in whether average speeds along the 100 miles might be different between the SUVs and the sedans. Unbeknownst to the drivers, all the motor vehicles were equipped with devices monitoring speed, acceleration, and other variables. The experiment yielded the following data:

SUV Speed

Sedan Speed

mean

69.95mph

61.58mph

sd

6.19mph

7.25mph

n

29

29

(i)

Calculate a 99 percent confidence interval for the difference between the mean speeds of the SUVs and the sedans.

(ii)

How do we interpret the results?

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

a.

(i)

(3.19, 13.15)

(ii)

We are 99 percent confident that among the men of the Motorist Society, the average difference in speeds between those driving SUVs and those driving sedans is 3.19 to 13.15 miles per hour.

b.

(i)

(3.19, 13.15)

(ii)

We are 99 percent confident that among drivers of SUVs and sedans, the average difference in speed between the SUVs and the sedans is 3.19 to 13.15 miles per hour.

c.
(i)
(3.53, 12.81) (ii)

We are 99 percent confident that among drivers of SUVs and sedans, the average difference in speed between the SUVs and the sedans is 3.53 to 12.81 miles per hour.

d.

(i)

(3.53, 12.81)

(ii)

We are 99 percent confident that among the men of the Motorist Society, the average difference in speeds between those driving SUVs and those driving sedans is 3.53 to 12.81 miles per hour.

e. None of the above answers sets is correct.

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Answer #1

(i)

Form given information, 5380.0.65.0.35 = 1223.95 and 6810.0.70.0.30 = 1430.1 both are greater than . Samples are independent samples. Since population is large so the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes. Hence, conditions are fulfilled.

(ii)

Here we have m =5380, Ê =0.65 n =6810,2 =0.7 The standard error is: P. (1-ê), P2(1-P2) -0.0086 SE=1 V 17 Level of significanc

Correct option is:

a. (i) The product of 5,380 and 65 percent must equal or surpass 10, as must the product of 5,380 and 0.35; samples from BC and PEC must be selected independently; the sizes of the samples of system-using commuters must fall below 10 percent of the population sizes; and samples selection must be randomized. (ii) (-0.067, -0.033)

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