Practice Test 5 - Statistics and Probability

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

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1. A bicycle shop models the cost of renting a bike with $y = 9x + 12,$ where $x$ is the number of hours rented (hours) and $y$ is the total rental cost (dollars). Which interpretation of the $y$-intercept is correct? (Include units.)
2. School context: Event A is "a student is in the robotics club," and event B is "a student is in the band." The principal says, "Robotics club members are more likely to be in band." Which comparison best explains conditional probability here?
3. A cafeteria offers 8 equally likely lunch combos, listed below: 1) Sandwich + Milk 2) Sandwich + Water 3) Sandwich + Juice 4) Salad + Milk 5) Salad + Water 6) Salad + Juice 7) Pasta + Milk 8) Pasta + Water A student randomly selects one combo from the list. Let event $A$ be "the drink is Water" and event $B$ be "the food is Salad." Using the outcomes listed, what is $P(A\mid B)$? Give your answer as a simplified fraction.
4. Using the outcomes listed in $S=\{\text{small},\text{medium},\text{large}\}$ for a randomly selected T-shirt size, event $A$ is the event that the size is not medium. Which set of outcomes represents event $A$?
5. A report states that neighborhoods with higher average home prices tend to have lower average commute times (a negative association), based on public records. The report compared neighborhoods as they are and did not assign home prices or commute times. Which statement is the most reasonable conclusion?
6. A company tracked the number of customer support tickets received each day for 12 days: 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 16, 31 How does the outlier (31 tickets) affect the spread of the data?
7. A used-car website fits a line to predict price from age: $y = -1500x + 22000,$ where $x$ is the car's age (years) and $y$ is the predicted price (dollars). What does the slope represent in this context? (Include units.)
8. A school store advertises: "Students prefer Brand X water bottles 3 to 1 over Brand Y." The store's data come from sales last month: 300 Brand X bottles sold and 100 Brand Y bottles sold. The ad does not mention that Brand X was priced at $8 while Brand Y was priced at $14. Which critique best evaluates the claim?
9. A board game has two sequential outcomes. First, you roll a 6 (event $A$). Given that you rolled a 6, you then draw a "bonus" card from a deck where 3 out of 10 cards are bonus cards (event $B$). If $P(A)=\frac{1}{6}$ and $P(B\mid A)=\frac{3}{10}$, what is $P(A\cap B)$, the probability that both $A$ and $B$ occur?
10. A school collected data from 100 students about whether they ride the bus or get a ride to school, and whether they arrived on time. The two-way table shows the counts. Which value represents the joint relative frequency (as a percent) of a student who rides the bus and arrives on time?

    	On time	Late	Total
    Rides the bus	48	12	60
    Gets a ride	36	4	40
    Total	84	16	100

11. A website claims that 10% of visitors click an ad: $P(\text{click})=0.10$. In a random sample of 60 visitors, 16 clicked. A student simulates 200 samples of 60 visitors under $P(\text{click})=0.10$; only 1 of the 200 simulated samples had 16 or more clicks. Would this result cause you to question the model? Why? (Treat "as extreme or more extreme" as $\ge 16$ clicks out of 60.)
12. A student recorded paired data for daily screen time (hours) and hours spent reading for 8 days. Technology reports a correlation coefficient of $r=-0.30$. Which statement best interprets the value of $r$ for the relationship between daily screen time and hours spent reading?
13. A quality control process checks a lightbulb in two stages. First, it must pass a brightness test (event $A$). Then, if it passed brightness, it must pass a lifespan stress test (event $B$). Suppose $P(A)=0.75$ and $P(B\mid A)=0.80$. What is the probability that both $A$ and $B$ occur?
14. A lab follows a two-step procedure. First, a sample is chosen from Batch 1 with probability $P(A)=0.30$. Then, given it came from Batch 1, the probability it contains contamination is $P(B\mid A)=0.05$. What is the probability that both events occur, $P(A\cap B)$?
15. A survey company claims that 50% of people prefer Brand A, so $P(\text{prefer A})=0.50$. In a random sample of 80 people, 62 prefer Brand A. A student simulates 100 samples of size 80 using $P(\text{prefer A})=0.50$ and checks for results 62 or more preferring A. In the simulations, 0 out of 100 samples had 62 or more. Which conclusion is most reasonable?
16. Adult male heights in a large population are approximately normal. Suppose heights are modeled as $N(\mu=70\text{ in},\,\sigma=3\text{ in})$. Approximately what percent of adult males are below 67 inches? (Use the empirical rule.)
17. A teacher must pick 1 of 4 topics (A, B, C, D) for a review session and wants a fair random choice, meaning each topic has probability $\tfrac14$. Which procedure is fair?
18. A student can choose one of two ways to earn points on a practice quiz each day. Strategy A: Guaranteed 7 points. Strategy B: With probability 0.50 you earn 20 points; with probability 0.50 you earn 0 points. Over many repetitions, which strategy has the greater expected value (average points per day)?
19. A teacher collected the number of pages read by 18 students over a weekend: 22, 24, 25, 25, 26, 27, 28, 28, 29, 30, 30, 31, 32, 34, 35, 36, 38, 40. Which statement correctly describes the distribution shown by a box plot of these data?
20. A factory uses an automated scanner to flag defective parts before shipping. Historically, about $2\%$ of parts are truly defective. The scanner has: - Sensitivity: $90\%$ (it flags $90\%$ of truly defective parts) - Specificity: $95\%$ (it correctly clears $95\%$ of truly good parts) If a part is flagged, the factory can choose one of these strategies: 1) Ship it anyway. 2) Do a manual inspection on flagged parts only. 3) Scrap all flagged parts with no further checks. The factory's goal is to minimize the chance that a defective part gets shipped, even if that means extra inspections or scrapping some good parts. Which strategy is most reasonable given the probabilities?

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