Practice Test 7 - Statistics and Probability

19 questions. Answer them in any order, then submit to see your score and every explanation.

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1. A simple wager game works like this: the player pays $1 to flip a fair coin. If it lands heads, the player receives $3. If it lands tails, the player receives $0. What is the expected value (long-run average net gain/loss) for one play, including the $1 cost?
2. A warehouse is choosing between two barcode scanners for order picking. Each scanned item is either correct (right product) or incorrect (wrong product). Based on trials: - Scanner 1: correctly identifies the right product $98\%$ of the time, but when it misreads, it fails to alert the worker $70\%$ of the time. - Scanner 2: correctly identifies the right product $96\%$ of the time, but when it misreads, it fails to alert the worker only $20\%$ of the time. The warehouse's goal is to minimize the chance that a wrong product is shipped (a wrong shipment happens when the scanner misreads and fails to alert). Which strategy is most reasonable given the probabilities?
3. A die is rolled once. Let event $A$ be "the roll is a multiple of 3" and event $B$ be "the roll is even." Given that $P(A)=\tfrac{1}{3}$, $P(B)=0.5$, and $P(A\cap B)=\tfrac{1}{6}$, are $A$ and $B$ independent? Justify using probabilities.
4. A website tracks visits. Let event $A$ be "a visit is from a mobile device" and event $B$ be "a visit results in a purchase." Given that $P(A)=0.60$, $P(B)=0.20$, and $P(A\cap B)=0.15$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
5. A music teacher randomly assigned 44 beginners to learn a short piece using either Treatment A (practice with a metronome) or Treatment B (practice without a metronome), 22 students per group. The outcome was performance accuracy (percent of notes played correctly). Results: Treatment A mean = 91%, Treatment B mean = 84%. Observed difference in means (A − B) = $7$ percentage points. A randomization test was performed by shuffling the treatment labels 1,000 times under "no treatment effect." In the simulation, 12 of the 1,000 shuffled differences were at least as large as $7$ percentage points. Do the results provide evidence that Treatment A is more effective than Treatment B?
6. A teacher wants to estimate the population parameter $\mu$, the mean number of hours of sleep students at the school get on a typical school night. A random sample of $n=50$ students had a sample statistic $\bar{x}=7.1$ hours. Students used simulation (4,000 repeated random samples of size 50) and found that about 95% of simulated sample means were within 0.6 hours of 7.1. Which interval is a reasonable estimate range for the population parameter $\mu$? (Round to the nearest tenth.)
7. A prize wheel has 5 equal sections labeled $0, $0, $1, $2, and $5. You spin once. Let $X$ be the amount of money won. Which table correctly represents the probability distribution of $X$?​
8. A snack machine is tested for 50 purchases. The observed outcomes were: no snack dispensed (0 snacks) occurred 4 times, 1 snack occurred 43 times, and 2 snacks occurred 3 times. Let $X$ be the number of snacks dispensed per purchase. Using the observed relative frequencies as estimated probabilities, what is the expected value of $X$ based on the empirical distribution?
9. A utility company analyzed last year's records and found that on days with higher outdoor temperatures, the average household electricity use (in kWh) was higher (a positive association). This was based on existing billing and weather records, not a controlled experiment. Which statement is the most reasonable conclusion?
10. School context: Two events are defined. Event A: a student brought a laptop to class today. Event B: a student sat in the front half of the classroom today. The teacher says, "Among students sitting in the front half, 30% brought a laptop." Which statement best describes what "the chance of A given that B" means in this situation?
11. A city's water department models a household's monthly water bill with $y = 18 + 2.5x,$ where $x$ is the number of thousands of gallons of water used in a month (thousand gallons) and $y$ is the monthly bill (dollars). What does the slope represent in this context? Include units in your interpretation.
12. At a high school, students were surveyed about whether they participate in an after-school club and whether they prefer studying in the morning or evening. The results are shown in the two-way table. What is the conditional relative frequency (as a percent) of students who prefer studying in the morning given that the student participates in an after-school club?

    	Prefer Morning	Prefer Evening	Total
    Club: Yes	28	52	80
    Club: No	40	30	70
    Total	68	82	150

13. A music instructor randomly assigned 48 beginners to two practice strategies for learning a short piece: Treatment A (practice with a metronome) and Treatment B (practice without a metronome). Twenty-four students were randomly assigned to each strategy. After one week, the mean performance rating (0–10 scale) was 7.3 for Treatment A and 6.5 for Treatment B, so the observed difference in means was $7.3-6.5=0.8$ (A − B). The instructor performed a randomization test by shuffling the strategy labels 2000 times and recalculating (A − B). In the 2000 shuffles, 14 simulated differences were $0.8$. Which conclusion is most reasonable about the treatment effect?
14. A school cafeteria randomly assigned 90 students to taste-test two smoothie recipes: Treatment A (adds banana) and Treatment B (no banana). Forty-five students were randomly assigned to each recipe. Students rated the smoothie from 1 to 10. Treatment A had a mean rating of 7.6 and Treatment B had a mean rating of 6.9, so the observed difference in means was $7.6-6.9=0.7$ (A − B). A randomization test was run by shuffling the recipe labels 3000 times and computing the difference in means each time. In the simulation distribution, 2 of the 3000 shuffled differences were at least as large as 0.7 (A − B). Do the results provide evidence that Treatment A is more effective than Treatment B (higher mean rating)?
15. A pet store chain wants to estimate the population parameter $p$, the proportion of all customers who buy pet food at least once per month. A random sample of $n=250$ customers gave a sample statistic $\hat{p}=0.72$. A simulation of repeated random sampling (2,500 samples of size 250) produced a distribution of simulated sample proportions. The store summarized the simulation: "About 95% of simulated $\hat{p}$ values were between 0.68 and 0.76." Which interval is a reasonable estimate range for the population parameter $p$? (Answer in percent.)
16. A runner recorded the pace (minutes per mile) for 14 training miles: 8.2, 8.3, 8.3, 8.4, 8.4, 8.5, 8.5, 8.6, 8.6, 8.7, 8.7, 8.8, 8.9, 11.5. The 11.5-minute pace occurred on a day with strong headwinds. How does this outlier most likely affect the mean and the median pace?
17. A teacher made a scatter plot of (x) the day of the month (1–30) and (y) the number of minutes a student spent reading that day. The points form a curved pattern: reading time is high at the beginning and end of the month but lower in the middle. Technology reports $r\approx 0.02$. Which statement best interprets the value of $r$?
18. A large set of exam scores is described as roughly symmetric and bell-shaped, so a normal model is appropriate. Assume scores are normally distributed with mean $\mu=75$ and standard deviation $\sigma=8$. Approximately what percent of students score between 67 and 83? (Use the empirical rule.)
19. A band director recorded the lengths (in minutes) of 13 practice sessions: 18, 20, 20, 22, 24, 25, 25, 25, 26, 28, 30, 30, 32. Which statement correctly describes the distribution shown by a dot plot?

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