Free-response practice · Question 3 style · 10 points

Road-test pass rates at two centers

A state's driver licensing agency runs two road-test centers, North and South. Last year, North gave 5,120 first-attempt road tests and South gave 4,380. An agency analyst selected a random sample of 240 first-attempt road tests from North's records and, independently, a random sample of 260 first-attempt road tests from South's records. The results are shown in the table.

First-attempt road tests at the two centers
Center	First-attempt tests given last year	Tests in the sample	Sampled tests that were passed
North	5,120	240	163
South	4,380	260	148

The analyst wants to estimate, with 95% confidence, the difference between the proportions of all first-attempt road tests given last year that were passed at North and at South.

Use the given information to respond to parts A, B, C, D, and E. Label any subparts (e.g., i and ii) that may be present.

A.

Identify the appropriate inference procedure.
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Score part A
- 1 point Identifies the procedure as a two-sample z-interval for a difference between two population proportions. The correct formula, $({\hat{p}}_{N} - {\hat{p}}_{S}) \pm z^{*} \sqrt{\frac{{\hat{p}}_{N} (1 - {\hat{p}}_{N})}{n_{N}} + \frac{{\hat{p}}_{S} (1 - {\hat{p}}_{S})}{n_{S}}}$ , also earns the point.Earns it: "two-sample z-interval for a difference between two population proportions"Does not: "two-sample z-test for a difference between two population proportions" (the analyst wants an estimate, so the procedure is an interval); "two-sample t-interval for a difference between two population means" (each test is passed or failed, so the parameters are proportions); "one-sample z-interval for a population proportion" (there are two populations to compare)
Model response
A two-sample z-interval for a difference between two population proportions.
B.

Complete the inference procedure, including calculating the appropriate interval.
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Score part B
- 1 point Parameter: defines the parameter as $p_{N} - p_{S}$ , where $p_{N}$ is the proportion of all first-attempt road tests given at North last year that were passed and $p_{S}$ is the same proportion for South. Both populations must be identified.Earns it: " $p_{N} - p_{S}$ , where $p_{N}$ and $p_{S}$ are the proportions of all first-attempt road tests passed at North and at South last year"Does not: " ${\hat{p}}_{N} - {\hat{p}}_{S}$ , the difference in the sample proportions" (that is the statistic, about 0.110, not the parameter); " $p$ = the proportion of drivers who pass" (no difference and no populations)
- 1 point Randomization and 10% conditions: states that the data come from two independent random samples, one from each center's records, and that each sample is no more than 10% of its population ( $240 \leq 512$ , which is 10% of 5,120, and $260 \leq 438$ , which is 10% of 4,380). Both conditions are needed for this point.Earns it: "Two independent random samples. 240 is less than 10% of 5,120 (512), and 260 is less than 10% of 4,380 (438)."Does not: "Random check: met." (no justification); "The samples were random." (the 10% condition is not checked); "240 and 260 are both at least 30." (not a condition for this procedure)
- 1 point Normality condition: states that the condition is met because the observed numbers of successes and failures in both samples are at least 10: 163 and 77 at North, and 148 and 112 at South.Earns it: "Successes and failures: 163 and 77 at North, 148 and 112 at South, all at least 10."Does not: " $n \geq 30$ for both samples" (sample size alone does not check the normality condition for proportions); " $n p \geq 10$ and $n (1 - p) \geq 10$ " (no values are shown for either sample); "By the central limit theorem, the sampling distribution is normal." (for proportions, the condition is checked with the counts of successes and failures)
- 1 point Interval: calculates $(0.679 - 0.569) \pm 1.96 \sqrt{\frac{0.679 (0.321)}{240} + \frac{0.569 (0.431)}{260}} = 0.110 \pm 0.084$ , or about $(0.026, 0.194)$ . Minor rounding differences are accepted, and an arithmetic or transcription error can be ignored if correct work is shown.Earns it: " $0.110 \pm 1.96 (0.0430) = (0.026, 0.194)$ "; "(0.026, 0.194) for North minus South"Does not: "(0.039, 0.181)" (uses $z^{*} = 1.645$ , the critical value for 90% confidence); " $0.110 \pm 0.043$ " (leaves out the critical value)
Model response
Let $p_{N}$ be the proportion of all first-attempt road tests given at North last year that were passed, and let $p_{S}$ be the same proportion for South. The parameter is $p_{N} - p_{S}$ .

The data come from two independent random samples of the centers' records, so the randomization condition is met. The 10% condition is met because 240 is less than 10% of 5,120 (512) and 260 is less than 10% of 4,380 (438). The normality condition is met because the observed numbers of successes and failures, 163 and 77 at North and 148 and 112 at South, are all at least 10.

${\hat{p}}_{N} = \frac{163}{240} \approx 0.679$ and ${\hat{p}}_{S} = \frac{148}{260} \approx 0.569$ .

$(0.679 - 0.569) \pm 1.96 \sqrt{\frac{0.679 (0.321)}{240} + \frac{0.569 (0.431)}{260}} = 0.110 \pm 0.084$ , so the 95% confidence interval is about $(0.026, 0.194)$ .
C.

Interpret the interval from part B in context.
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Score part C
- 1 point Includes the 95% confidence level and the interval endpoints, 0.026 and 0.194, or endpoints consistent with the response to part B.Earns it: "We are 95% confident that the interval from 0.026 to 0.194 ..."Does not: "The interval is (0.026, 0.194)." (no confidence level); "We are 95% confident." (no endpoints)
- 1 point Includes the context: the interval estimates the difference (North minus South) between the true proportions of all first-attempt road tests passed last year at North and at South.Earns it: "... captures the difference between the proportions of all first-attempt road tests passed at North and at South last year (North minus South)."Does not: "... contains the difference in pass rates." (does not say whose pass rates or that the parameter is about all tests at the two centers); "... contains the difference between the sample proportions." (the sample difference is known exactly; the interval estimates the population difference)
Model response
We are 95% confident that the interval from 0.026 to 0.194 captures the difference (North minus South) between the proportions of all first-attempt road tests passed at North and at South last year.
D.

Explain what is meant by 95% confidence in this context.
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Score part D
- 1 point Explains the confidence level as the long-run capture rate of the method: if many pairs of random samples of the same sizes were taken from the two centers' records and a 95% interval were constructed from each pair, about 95% of the intervals would capture the true difference between the proportions of first-attempt road tests passed at North and at South.Earns it: "If we took many random samples of 240 North tests and 260 South tests and built a 95% interval each time, about 95% of the intervals would capture the true difference in the proportions passed."Does not: "There is a 95% probability that the true difference is between 0.026 and 0.194." (this interval either captures the true difference or it does not; the 95% describes the method); "95% of first-attempt road tests are passed." (confuses the confidence level with a pass rate); "95% of sample differences fall between 0.026 and 0.194." (the confidence level describes how often intervals capture the true difference, not where sample differences fall)
Model response
If the analyst repeatedly took independent random samples of 240 first-attempt tests from North and 260 from South and built a 95% confidence interval from each pair of samples, about 95% of those intervals would capture the true difference between the proportions of first-attempt road tests passed at North and at South last year.
E.

i. Based on the interval, determine whether there is convincing statistical evidence that the proportions of first-attempt road tests passed last year at the two centers were different. Explain your reasoning.

ii. The agency's director claims that North's first-attempt pass rate last year was more than 10 percentage points higher than South's. Determine whether the interval supports the director's claim. Explain your reasoning.
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Score part E
- 1 point Determines that there is convincing statistical evidence of a difference because 0 is not in the interval (every plausible value is positive, so North's proportion appears higher). The determination must be consistent with the interval in part B and must use the interval.Earns it: "Yes. 0 is not in the interval (0.026, 0.194), so there is convincing evidence that the proportions differ."Does not: "Yes, because 0.679 is greater than 0.569." (compares the sample proportions instead of using the interval); "No, because the interval is wide." (what matters is whether 0 is in the interval, and it is not)
- 1 point Determines that the interval does not support the director's claim, because it includes plausible values of 0.10 or less (from 0.026 up to 0.10), so a difference of 10 percentage points or less cannot be ruled out.Earns it: "No. The interval includes values below 0.10, such as 0.05, so it does not give convincing evidence that the difference is more than 10 percentage points."Does not: "Yes, because the point estimate 0.110 is more than 0.10." (the point estimate alone ignores sampling variability); "Yes, because the upper endpoint, 0.194, is more than 0.10." (values below 0.10 are also in the interval and are just as plausible); "No, the interval shows the difference is less than 0.10." (the interval also contains values above 0.10, so it cannot show that)
Model response
i. Yes. Every value in the interval is positive, so 0 is not a plausible value for $p_{N} - p_{S}$ . There is convincing statistical evidence that the proportions of first-attempt road tests passed at the two centers were different, with North's proportion higher.

ii. No. The interval includes plausible values of 0.10 or less, such as 0.05, so a difference of 10 percentage points or less is plausible. The interval does not provide convincing statistical evidence that North's pass rate was more than 10 percentage points higher than South's.

Your score: 0 of 10. Each point is earned on its own, the way the exam scores it.