Practice Quiz with Solutions: Chapter 12 / MATH200 (TC3, Brown)

Practice Quiz with Solutions: Chapter 12 (21 min)

Before looking at these solutions, please work the practice quiz.

Disclaimer: This quiz is representative of the level of difficulty you should expect, but it doesn’t include every single topic from the week’s work. The real quiz may include some other topics and may skip some that are in this practice quiz. (The real quiz also may not word questions in the same way as the practice quiz. You should focus on the concepts, not a particular form of words.)

These solutions show about the same level of work I expect from you, though I often add some extra commentary. Please see Show Your Work for the what, why, and how.

1(points: 7) You think that people’s ice cream favorites are 25% each vanilla and chocolate, 20% strawberry, 15% butter pecan, 8% rocky road, and 7% other or no preference. You randomly survey 1000 people and find preferences are 220 vanilla, 255 chocolate. 190 strawberry, 170 butter pecan, 95 rocky road, and 70 other or no preference. Using α=0.05, was your idea right or wrong?

Solution: This is attribute data, one population, more than two responses: Case 6, goodnesss-of-fit. There are 6 categories, therefore 5 degrees of freedom.

(1)	H₀: The 25:25:20:15:8:7 model for ice cream preference is good. H₁: The 25:25:20:15:8:7 model for ice cream preference is bad.
(2)	α = 0.05
(3–4)	Use MATH200A Program part 6. Here are the input and output data screens: You will write something like MATH200A/GOF and then the values: χ²=9.68, df=5, pval = 0.0849 Common mistake: When a model is given in percentages, some students like to convert the observed numbers to percentages. Never do this! The observed numbers are always actual counts and their total is always the actual sample size. Alternative solution: You could give the model as decimals, .25, .20, .15 and so on. But for the model, all that matters is the relative size of each category to the others, so it’s simpler to use whole-number ratios. Common mistake: If you do convert the percentages to decimals, remember that 8% and 7% are 0.08 and 0.07, not 0.8 and 0.7.
(RC)	L3 shows the expected values, and they are all >5. The problem says that the 1000 people were a random sample.
(5)	p>α; fail to reject H₀.
(6)	At the 0.05 level of significance, you can’t say whether the model is good or bad. Remark: For Case 6 only, you could write your non-conclusion as something like “the model is not inconsistent with the data” or “the data don’t disprove the model.” Remark: Note that the χ² test keeps you from jumping to false conclusions. Eyeballing the observed and expected numbers (L2 and L3), you might think they’re fairly far off and the model must be wrong. Yet the test gives a largish p-value. Remark: If it had gone the other way — if p was less than α — you would say something like “At the .05 level of significance, the model is inconsistent with the data” or “the data disprove the model” or simply “the model is wrong”.

(1)

H₀: The 25:25:20:15:8:7 model for ice cream preference is good.
H₁: The 25:25:20:15:8:7 model for ice cream preference is bad.

(2)

α = 0.05

(3–4)

Use MATH200A Program part 6. Here are the input and output data screens:

TI-83 lists, per text TI-83 results screen, per text

You will write something like MATH200A/GOF and then the values: χ²=9.68, df=5, pval = 0.0849

Common mistake: When a model is given in percentages, some students like to convert the observed numbers to percentages. Never do this! The observed numbers are always actual counts and their total is always the actual sample size.

Alternative solution: You could give the model as decimals, .25, .20, .15 and so on. But for the model, all that matters is the relative size of each category to the others, so it’s simpler to use whole-number ratios.

Common mistake: If you do convert the percentages to decimals, remember that 8% and 7% are 0.08 and 0.07, not 0.8 and 0.7.

(RC)

TI-83 lists with expected values in L3, per text

L3 shows the expected values, and they are all >5. The problem says that the 1000 people were a random sample.

(5)

p>α; fail to reject H₀.

(6)

At the 0.05 level of significance, you can’t say whether the model is good or bad.

Remark: For Case 6 only, you could write your non-conclusion as something like “the model is not inconsistent with the data” or “the data don’t disprove the model.”

Remark: Note that the χ² test keeps you from jumping to false conclusions. Eyeballing the observed and expected numbers (L2 and L3), you might think they’re fairly far off and the model must be wrong. Yet the test gives a largish p-value.

Remark: If it had gone the other way — if p was less than α — you would say something like “At the .05 level of significance, the model is inconsistent with the data” or “the data disprove the model” or simply “the model is wrong”.

	favor	oppose	unsure	total
Democrat	440	400	120	960
Republican	320	480	100	900
total	760	880	220	1860

2(points: 7) Democrats and Republicans were surveyed for their opinions on gun control, and the results are shown in the table at right. Based on this sample, does a person’s opinion on gun control depend on party affiliation, at the .05 level of significance?

Solution: Use Case 7, 2-way table.

(1)	H₀: Gun opinion is independent of party H₁: Gun opinion depends on party
(2)	α = .05
(3–4)	Put the two rows and three columns in matrix A. (Don’t enter the totals.) Select χ²-Test from the menu. Outputs are χ² = 26.13, df = 2, p = 2.1181E-6 or 0.000 002 or <.0001.
(RC)	Check the B matrix and verify that all expected values are well above the minimum of 5. Alternative: use MATH200A part 7 for steps 3–4 and RC.
(5)	p < α; reject H₀ and accept H₁.
(6)	At the .05 level of significance, gun opinion depends on party.

(1)

H₀: Gun opinion is independent of party
H₁: Gun opinion depends on party

(2)

α = .05

(3–4)

Put the two rows and three columns in matrix A. (Don’t enter the totals.) Select χ²-Test from the menu. Outputs are χ² = 26.13, df = 2, p = 2.1181E-6 or 0.000 002 or <.0001.

(RC)

Check the B matrix and verify that all expected values are well above the minimum of 5.

Alternative: use MATH200A part 7 for steps 3–4 and RC.

(5)

p < α; reject H₀ and accept H₁.

(6)

At the .05 level of significance, gun opinion depends on party.

3(points: 1) 425 school children were surveyed about what they want to be when they grow up, out of a choice of five professions. The results were Teacher 80, Doctor 105, Lawyer 70, Police officer 70, Firefighter 100.

Obviously these particular children preferred some occupations over others. You want to test whether their preferences reflect a real difference in the population. What model do you use? (Write only the model; don’t do a hypothesis test.)

Answer: 1:1:1:1:1

Alternative solution: Any other set of five equal numbers would be correct.

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