Practice Test 38 - Statistics and Probability

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

Start practice test

1. A random sample of 150 students recorded whether they prefer to study in the morning or evening and whether they usually study alone or in a group. Using the two-way table, what is $P(\text{Morning}\mid\text{Group})$? Give your answer as a fraction. Two-way table (counts): - Group & Morning: 18 - Group & Evening: 42 - Alone & Morning: 45 - Alone & Evening: 45 - Row totals: Group = 60, Alone = 90 - Column totals: Morning = 63, Evening = 87 - Grand total: 150
2. A teacher drops the class's lowest test score, an outlier. What happens?
3. A coach wants to know whether listening to music during warm-ups improves free-throw percentage. Players choose for themselves whether to listen to music during warm-ups for a week, and the coach records each player's free-throw percentage at the end of the week. Which type of study is described?
4. A machine produces metal rods whose lengths are approximately normal (roughly symmetric, bell-shaped) with mean $\mu = 50.0$ cm and standard deviation $\sigma = 1.0$ cm. Approximately what percent of rods have lengths between 49.0 cm and 52.0 cm? (Use the empirical rule.)
5. In a randomized experiment, 60 students were randomly assigned by shuffling identical cards (30 labeled A and 30 labeled B) to either Treatment A (a new study app) or Treatment B (a standard app). After one week, each student took the same 20-question quiz. The mean score for Treatment A was 16.4 and for Treatment B was 14.9, so the observed difference in means was $16.4-14.9=1.5$ points (A − B). To test whether this difference could be due to chance, the researcher performed a randomization test by keeping all 60 quiz scores fixed and repeatedly shuffling the A/B labels 2000 times under "no treatment effect." In the 2000 shuffles, 18 produced a difference in means of at least $+1.5$ points (A − B). Which conclusion is most reasonable about the treatment effect (A − B)?
6. A tech company is deciding how to handle login attempts that look suspicious. Internal data show: - On a typical day, $0.5\%$ of login attempts are truly malicious. - The detection system flags $92\%$ of malicious attempts. - It also falsely flags $4\%$ of legitimate attempts. They are choosing a policy for flagged attempts. The goal is to minimize the chance that a malicious attempt is allowed through, while avoiding treating every user as malicious. Which strategy is most reasonable given the probabilities? - Strategy 1: Allow flagged attempts but send a warning email. - Strategy 2: Require a second factor (2FA) only for flagged attempts. - Strategy 3: Block all flagged attempts immediately.
7. A pet food company wants to estimate the proportion of all customers who are satisfied with a new dog food formula. From the company's customer list, it randomly selects 120 customers to survey; 90 say they are satisfied. Why might the company still expect the sample result to differ from the true population proportion?
8. A survey asks a randomly selected student how many siblings they have. Suppose the only possible responses in this group are 0, 1, or 2 siblings, with probabilities $P(0)=0.30$, $P(1)=0.50$, and $P(2)=0.20$. Let $X$ be the number of siblings reported. Which graph represents the distribution of $X$?
9. A streaming service tracks preferences. Event $A$ is "a randomly selected user watches comedy," and event $B$ is "a randomly selected user watches documentaries." The data show $P(A)=0.44$, $P(B)=0.36$, and $P(A\cap B)=0.12$. What is the probability that $A$ or $B$ occurs, where "or" is inclusive (comedy or documentaries or both)?
10. A survey of 60 commuters shows that 24 ride the bus (event $B$). Of those who ride the bus, 9 also commute before 7 a.m. (event $A \cap B$), where event $A$ is "a commuter travels before 7 a.m." Based on the information given, what is $P(A \mid B)$? Give your answer as a simplified fraction.
11. Lottery ticket costs $4. Payouts: $1000 with probability .001, $20 with probability .01, otherwise $0. Find expected net gain.
12. A teacher uses the fitted line $y = 92 - 3x$ to model a student's quiz score, where $x$ is the number of questions missed (questions) and $y$ is the quiz score (points). Which interpretation of the slope is correct? Include units in your interpretation.
13. A spinner is divided into 5 equal sectors labeled $A,B,C,D,E$, so each outcome is equally likely. Define the random variable $X$ by the mapping: $A\mapsto 0$, $B\mapsto 0$, $C\mapsto 1$, $D\mapsto 2$, $E\mapsto 5$. What is the expected value of $X$?
14. A fair coin is flipped 4 times; all $16$ outcomes are equally likely. Define the random variable $X$ as the number of times two consecutive flips are both heads (counting overlaps). For example, $\text{HHHT}$ has two occurrences of consecutive heads (positions 1–2 and 2–3), so $X=2$. The mapping by outcome type includes: $\text{TTTT}\to 0$, $\text{HHTT}\to 1$, $\text{HHHT}\to 2$, $\text{HHHH}\to 3$, etc. What is the expected value of $X$ based on this theoretical model?
15. A community group records the number of minutes spent exercising per day (minutes) and the number of hours spent watching TV per day (hours) for several adults. A scatter plot shows a roughly linear downward trend, and technology reports $r=-0.54$. Which statement best interprets the value of $r$ for the relationship between exercise minutes and TV hours?
16. A game booth claims its prize wheel lands on Blue with probability $P(\text{Blue})=0.25$. You watch 40 spins and see Blue come up 22 times. A student runs 100 simulated sets of 40 spins using the model $P(\text{Blue})=0.25$; in those simulations, 0 out of 100 sets had 22 or more Blues. Would this result cause you to question the model? Why? (Treat "as extreme or more extreme" as getting $\ge 22$ Blues in 40 spins.)
17. The daily number of support tickets received over 11 days was: 31, 32, 33, 33, 34, 34, 35, 35, 36, 37, 60 Which measure is most affected by the outlier 60 tickets?
18. According to $S=\{1,2,3,4,5,6\}$ for rolling one standard number cube, event $A=\{x\in S\mid x\text{ is odd}\}$ and event $B=\{x\in S\mid x\ge 4\}$. Which set of outcomes represents event $B$?
19. A store tracked the number of minutes customers spent waiting in line on two different days. Data Set A (Day 1): 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 Data Set B (Day 2): 1, 2, 2, 3, 3, 4, 4, 6, 7, 9, 12, 18 Data Set A is roughly symmetric. Data Set B is right-skewed with a few large values. Which measures are most appropriate for comparing the typical value and variability of these distributions?
20. A community center wants to estimate the population parameter $p$, the proportion of all local residents who would attend a free weekend class. A random sample of 300 residents found that 147 said "yes," so the sample statistic is $\hat p = 147/300 = 0.49$ (49%). Students used simulation to model repeated random sampling of size 300. Their summary said: "In 1,000 simulated samples, about 95% of the simulated sample proportions were within 0.06 of 0.49." What does the margin of error mean in context?

Practice tests