Unit 3 · 15–25% of the exam Beta

Unit 3 practice test: Inference for Categorical Data: Proportions

12 exam-style questions from the unit, free, with an explanation for every option. Most have four answer choices, as on the real exam, and a few ask you to type a value, an interval or the setup; sets share one data set, plot or table.

Covers: Estimators; Sampling Distributions for Sample Proportions; Constructing a Confidence Interval for a Population Proportion; Justifying a Claim Based on a Confidence Interval for a Population Proportion; Setting Up a Test for a Population Proportion; p-Values; Carrying Out a Test for a Population Proportion; Potential Errors When Performing Tests; Sampling Distributions for the Difference Between Sample Proportions; Constructing a Confidence Interval for the Difference Between Two Population Proportions; Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Proportions; Setting Up a Test for the Difference Between Two Population Proportions; Carrying Out a Test for the Difference Between Two Population Proportions; Setting Up a Chi-Square Test for Homogeneity or Independence; Carrying Out a Chi-Square Test for Homogeneity or Independence.

Table

Getting to school at three high schools

A district takes a separate random sample of students at each of its three high schools and asks how they usually get to school. (Invented data.)

Observed counts: school by travel mode
School	Bus	Car	Walk or bike	Total
Central	60	55	35	150
East	40	60	20	120
West	52	48	30	130
Total	152	163	85	400

1. Which test fits the district's question of whether the travel mode distributions differ among the schools?

A chi-square test for homogeneity
A chi-square test for independence
Three one-proportion z tests
A two-sample z test for the bus proportions

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Several samples or groups, one categorical variable: homogeneity.

Correct: A chi-square test for homogeneity: Right: three separate random samples, one categorical variable compared across them.
A chi-square test for independence: Independence uses one sample classified by two variables. Here the district chose how many students to sample at each school.
Three one-proportion z tests: Separate tests for one category at a time would not compare the whole distributions at once.
A two-sample z test for the bus proportions: That compares two schools on one category. There are three schools and three categories.

2. Which null hypothesis fits this test?

The three schools have the same number of students
The distribution of travel mode is different at each school
Bus, car and walking are equally common at each of the schools
The travel mode distribution is the same at all three schools

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$H_0$ : the distributions are the same; $H_a$ : at least one differs.

The three schools have the same number of students: The test compares the travel mode distributions, not school sizes.
The distribution of travel mode is different at each school: That is the alternative: at least one school's distribution differs.
Bus, car and walking are equally common at each of the schools: Equal shares for each mode is not the claim. $H_0$ says the schools share one distribution, whatever it is.
Correct: The travel mode distribution is the same at all three schools: Right: homogeneity means "same distribution" in every population.

3. Which statement checks the 10% condition for this test?

Each expected count is at least 10% of its row total
At least 10% of each sample walks or bikes
Each sample is less than 10% of the students at its own school
The 400 students are less than 10% of the district's adults

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10%: each sample is less than 10% of its population when sampling without replacement.

Each expected count is at least 10% of its row total: That is not a condition. Expected counts are checked against 5.
At least 10% of each sample walks or bikes: That is not the condition. The 10% condition compares sample size with population size.
Correct: Each sample is less than 10% of the students at its own school: Right: with three samples drawn without replacement, check each against its own population.
The 400 students are less than 10% of the district's adults: The populations are the students at each school, not the district's adults.

4. Which of these changes to the table would make the chi-square statistic larger, with the totals kept the same?

Expected counts that equal the observed counts
Observed counts that sit closer to their expected counts
Listing the schools in a different order
Observed counts farther from their expected counts

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The statistic measures how far the observed counts are from what $H_0$ predicts.

Expected counts that equal the observed counts: Then each term is 0 and $\chi^2 = 0$ , the smallest it can be.
Observed counts that sit closer to their expected counts: Closer counts give smaller terms and a smaller $\chi^2$ .
Listing the schools in a different order: Reordering the rows changes no count, so $\chi^2$ is the same.
Correct: Observed counts farther from their expected counts: Right: $\chi^2$ adds up $\frac{(O - E)^2}{E}$ , so larger gaps between observed and expected give a larger value.

Table

One poll, three intervals

A city council selects a random sample of 600 residents, and 324 of them ( $\hat{p} = 0.54$ ) favor longer library hours. A computer builds three confidence intervals for the proportion of all residents who favor longer hours, at 90%, 95% and 99% confidence, and lists them in random order.

Three intervals from the same sample, levels not shown
Interval	Lower end	Upper end
Interval 1	0.500	0.580
Interval 2	0.488	0.592
Interval 3	0.507	0.573

5. Which interval was built with 99% confidence?

Interval 1
Interval 2
Interval 3
There is no way to tell, because they come from the same sample

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Higher confidence needs a larger critical value (2.576 for 99%), so the margin of error is largest. Interval 2 reaches farthest from 0.54 on each side (from 0.488 to 0.592).

Interval 1: Interval 1, from 0.500 to 0.580, is in the middle: it is the 95% interval.
Correct: Interval 2: Right: the largest margin of error comes from the largest critical value.
Interval 3: Interval 3 has the smallest margin of error, so it has the lowest confidence, 90%.
There is no way to tell, because they come from the same sample: Same sample, same $\hat{p}$ and standard error, but different critical values, so different margins of error.

6. What is the margin of error of Interval 1?

0.040
0.080
0.540
0.020

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The margin of error is half the distance between the ends: $\frac{0.580 - 0.500}{2}$ $= 0.040$ . Equivalently, it is the distance from $\hat{p} = 0.54$ to either end.

Correct: 0.040: Right: half the distance between the two ends.
0.080: That is the whole distance from one end to the other. The margin of error is half of it.
0.540: That is the center of the interval, the point estimate, not the margin of error.
0.020: That is the standard error. The margin of error is the standard error times $z^* = 1.96$ .

7. Suppose the council had surveyed 2,400 residents and still found $\hat{p} = 0.54$ . About what would the margin of error of the 95% interval be?

0.020
0.010
0.040
0.080

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The margin of error is $z^*\sqrt{\hat{p}(1 - \hat{p})/n}$ . With 4 times the sample size it is divided by $\sqrt{4} = 2$ : about $\frac{0.040}{2} = 0.020$ .

Correct: 0.020: Right: 4 times the sample size halves the margin of error.
0.010: The sample size is inside a square root, so 4 times the residents divides the margin by 2, not 4.
0.040: A larger sample gives a smaller margin of error; it does not stay the same.
0.080: More residents give a more precise estimate, so the margin of error shrinks.

8. For a new poll, the council wants a 95% margin of error of at most 0.02. Using 0.54 as its estimate of the proportion, what is the smallest sample size it needs?

2,385
2,386
2,401
1,681

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Solve $1.96\sqrt{\frac{0.54(0.46)}{n}} \le 0.02$ : $n \ge \left(\frac{1.96}{0.02}\right)^2(0.54)$ $(0.46)$ $= 2385.6$ . Round up to 2,386 residents (always round a sample size up).

2,385: The calculation gives 2,385.6, and a sample size is always rounded up, or the margin would be a bit over 0.02.
Correct: 2,386: Right: $\left(\frac{1.96}{0.02}\right)^2(0.54)(0.46)$ $= 2385.6$ , rounded up.
2,401: That uses 0.5 in place of 0.54. It is the safe choice when there is no estimate, but this question says to use 0.54.
1,681: That uses $z^* = 1.645$ , the critical value for 90% confidence. For 95%, use 1.96.

9. A biologist wants the largest wingspan among a park's 3,000 bats. She captures a random sample of 30 bats and uses the largest wingspan in the sample. Which describes this estimator?

Unbiased, because the sample was chosen at random
Biased: it tends to underestimate the largest wingspan
Biased: it tends to overestimate the largest wingspan
Unbiased, because a sample of 30 bats is large enough

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The sample maximum can equal the population maximum but can never exceed it. So its sampling distribution sits at or below the true value: it tends to underestimate, which is bias.

Unbiased, because the sample was chosen at random: Random selection does not make every statistic unbiased. The sample maximum cannot go above the true maximum, so it tends to fall short.
Correct: Biased: it tends to underestimate the largest wingspan: Right: no sample can show a wingspan larger than the largest in the park, and most samples miss the largest bat.
Biased: it tends to overestimate the largest wingspan: A sample cannot contain a bat larger than the largest bat in the park, so it cannot overestimate.
Unbiased, because a sample of 30 bats is large enough: A larger sample would reduce the shortfall, but the sample maximum still tends to be too small.

10. In a large city, 50% of voters favor a new transit tax. A random sample of 400 voters is taken. Find the probability that 55% or more of the sample favors the tax.

Type a number. A table answer and a calculator answer are both accepted.

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Large counts: $400(0.5) = 200 \ge 10$ twice. $\sigma_{\hat{p}}$ $= \sqrt{\frac{0.5(0.5)}{400}}$ $= 0.025$ , so $z$ $= \frac{0.55 - 0.50}{0.025}$ $= 2.00$ and $P(\hat{p} \ge 0.55)$ $= P(Z \ge 2.00)$ $= 0.0228$ .

Answer: 0.0228

11. A meal-kit company says that 25% of adults in the state have used its type of service. Based on the interval from 0.214 to 0.262, what can you conclude?

0.25 is plausible: no convincing evidence against the company's figure
The company is right, because 0.25 is inside the interval
The company is wrong, because the sample proportion, 0.238, is not 0.25
The company is wrong, because 0.25 is not the center of the interval

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0.25 is inside the interval, so it is a plausible value for the true proportion. The interval does not give convincing evidence that the company is wrong.

Correct: 0.25 is plausible: no convincing evidence against the company's figure: Right: a value inside the interval cannot be ruled out.
The company is right, because 0.25 is inside the interval: Plausible is not proven; values such as 0.22 are plausible too.
The company is wrong, because the sample proportion, 0.238, is not 0.25: A sample proportion is expected to differ a little from $p$ ; the interval accounts for that.
The company is wrong, because 0.25 is not the center of the interval: The center is the sample proportion. Any value inside the interval is plausible.

12. A clinic says that more than half of its patients get an appointment within a week. A reporter wants to find evidence for the clinic's claim. Which hypotheses fit, where $p$ is the true proportion of all the clinic's patients seen within a week?

$H_0: p = 0.5$ , $H_a: p > 0.5$
$H_0: p > 0.5$ , $H_a: p = 0.5$
$H_0: p = 0.5$ , $H_a: p < 0.5$
$H_0: p = 0.5$ , $H_a: p \ne 0.5$

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The claim the reporter wants evidence for goes in the alternative: $H_a: p > 0.5$ , with $H_0: p = 0.5$ .

Correct: $H_0: p = 0.5$ , $H_a: p > 0.5$ : Right: evidence for "more than half" means $H_a: p > 0.5$ .
$H_0: p > 0.5$ , $H_a: p = 0.5$ : Reversed. A test looks for evidence for $H_a$ , so the claim to support goes there.
$H_0: p = 0.5$ , $H_a: p < 0.5$ : That would look for evidence that fewer than half are seen within a week.
$H_0: p = 0.5$ , $H_a: p \ne 0.5$ : The claim has a direction, "more than half", so the alternative is one-sided.

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Practice has every question in the unit, with feedback after each one; the notes for every topic are free. Other units: Unit 1 · Unit 2 · Unit 3 · Unit 4 · Unit 5.