Practice Test 15 - 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 university wants to estimate the proportion of all students currently enrolled who own a bicycle. It randomly selects 100 students from the enrollment list and finds that 34 own a bicycle. Which inference is supported by this random sample?
2. School context: Event A is "a student volunteers at least once this semester," and event B is "a student is in the student council." A teacher reports: "Of those in student council, 30% volunteer at least once." Which statement best describes what "the chance of A given B" means here?
3. A local tutoring company posts: "Our program raises math test scores by 20 points on average." They report that 12 students who signed up for tutoring had an average score of 60 on a pre-test and 80 on a post-test after 6 weeks. There was no comparison group of students who did not receive tutoring. Which statement best describes whether the conclusion is justified?
4. A city wants to estimate the population parameter $p$, the proportion of all households in the city that recycle weekly. A random sample of $n=200$ households found a sample statistic $\hat{p}=0.62$ (62%). Students used a simulation that repeatedly took 1,000 random samples of size 200 (assuming the true proportion is close to 0.62) and recorded the simulated sample proportions. In the simulation, about 95% of the simulated sample proportions were between 0.55 and 0.69. Which interval is a reasonable estimate range for the population parameter $p$ based on the simulation?
5. A fair spinner has 5 equal sections labeled A, B, C, D, and E. It is spun once. Define the random variable $X$ as the number of vowels spun, where vowels are A and E. Which table correctly represents the probability distribution of $X$?
6. School context: Two events are defined. Event A: a student joined the chess club. Event B: a student is in 10th grade. The club advisor says, "Being in 10th grade doesn't change the chance of joining chess club." Which statement best describes whether A and B are independent?
7. A taxi fare in a city is modeled by $y = 2.75x + 3.50,$ where $x$ is the distance traveled (miles) and $y$ is the total fare (dollars). What does the y-intercept represent in this context? (Include units.)
8. A company is choosing between two quality-control tests to run on many batches. Payoffs are net dollars to the company per batch. Strategy A: Test costs $3,000. Outcomes: with probability 0.95 the batch passes and there is no further cost (net payoff $-3,000); with probability 0.05 the test catches a defect and prevents a $100,000 recall (net payoff $100,000-3,000=$97,000). Strategy B: Test costs $6,000. Outcomes: with probability 0.98 the batch passes (net payoff $-6,000); with probability 0.02 the test catches a defect and prevents the $100,000 recall (net payoff $94,000). Which strategy would result in a higher average net payoff over many repetitions?
9. A scientist compared the amount of fertilizer applied to a plant (grams) with the plant's height after 6 weeks (cm) for several plants. Technology reported a correlation coefficient of $r = 0.88$. Which statement best interprets the value of $r$ for the relationship between fertilizer amount and plant height?
10. Based on the sample space shown, $S=\{\text{cat},\text{dog},\text{fish},\text{bird}\}$, one animal is selected at random. Event $A$ is the event that the animal can fly. Event $B$ is the event that the animal lives in water. Which expression represents $A\cup B$ (A or B)?
11. A spinner game defines a random variable $X$ as the number of points you score in one spin. The distribution is: - $P(X=1)=\tfrac{1}{3}$ - $P(X=4)=\tfrac{1}{3}$ - $P(X=7)=\tfrac{1}{3}$ What is the expected value of $X$ (in points)? (Note: the expected value may not be a possible outcome.)
12. A machine is supposed to produce a defect with probability $P(\text{defect})=0.05$ on each item. In a batch of 80 items, you observe 9 defects. A student simulates 120 batches of 80 items under the model; 2 out of 120 simulated batches had 9 or more defects. Would this result cause you to question the model? Why? (Treat "as extreme or more extreme" as $\ge 9$ defects out of 80.)
13. Four teams—Red, Blue, Green, and Yellow—are tied and need a fair way to decide who gets first pick. Fair means each team has probability $\tfrac{1}{4}$ of getting first pick. Which method gives each team an equal chance?
14. A bakery tests a new oven and records the internal temperature of a loaf at evenly spaced times while heating. The data are shown below. Time (minutes): 0, 2, 4, 6, 8, 10 Temperature (°C): 22, 32, 42, 52, 62, 72 Which type of function best models the relationship between time and temperature?
15. A two-step hiring process works as follows: first, an applicant passes the resume screen with probability $P(A)=0.40$. Then, given the applicant passes the resume screen, they pass the interview with probability $P(B\mid A)=0.25$. What is the probability that both the applicant passes the resume screen and passes the interview (i.e., $P(A\cap B)$)?
16. A teacher wants to estimate the population parameter $\mu$, the mean score (out of 100) on a district-wide quiz. A random sample of $n=50$ students had a sample statistic $\bar{x}=72$. Students used simulation by repeatedly taking 2,000 random samples of size 50 and recording the sample mean. They reported: about 95% of the simulated sample means were between 68 and 76. Which interval is a reasonable estimate range for the population parameter $\mu$?
17. A school wants to estimate the population parameter $p$, the proportion of all students at the school who usually bring a reusable water bottle. A random sample of 200 students found that 118 brought a reusable bottle, so the sample statistic is $\hat p = 118/200 = 0.59$ (59%). To model random sampling variability, students ran a simulation of 1,000 random samples of size 200 from a population where the proportion is unknown, but centered at the observed sample statistic. In the simulation, about 95% of the simulated sample proportions were between 0.53 and 0.65. Which interval is a reasonable estimate range for the population parameter $p$ based on the simulation?
18. From 10 raffle tickets labeled 1–10, 3 different winning tickets are drawn without replacement. The prizes are 1st, 2nd, and 3rd, so order matters. What is the probability that ticket 7 wins 1st prize and ticket 8 wins 2nd prize (the 3rd prize can be any other ticket)?
19. A student council flyer claims: "Most students want longer lunch." The council posted a poll link on its own social media account and received 310 responses. Of those, 240 said "Yes, longer lunch" and 70 said "No." The flyer concludes: "About 77% of students want longer lunch." Which critique best evaluates the claim?
20. A shipping company has a two-step process. First, a package is labeled "fragile" with probability $P(A)=0.35$. Then, given that a package is labeled fragile, it is handled with extra padding with probability $P(B\mid A)=0.60$. What is the probability that a package is labeled fragile and then handled with extra padding, $P(A\cap B)$?

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