Practice Test 44 - 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 freelancer must choose between two payment plans for the same type of job. Compare the expected payment per job over many jobs. Strategy A (Guaranteed + bonus chance): - With probability 0.70, get $120 - With probability 0.30, get $180 Strategy B (Higher upside, lower typical pay): - With probability 0.20, get $300 - With probability 0.80, get $110 Which strategy has the greater expected value (average payment) per job over many jobs?
2. A random sample of 140 students recorded whether each student rides the bus to school and whether they arrive before the first bell. Using the two-way table, are the events A = "rides the bus" and B = "arrives before the first bell" independent? Use the table by comparing $P(A\mid B)$ to $P(A)$. Two-way table (counts): - Rows: Rides the bus (Yes, No) - Columns: Arrives before bell (Yes, No) Bus Yes: Before Yes 30, Before No 40 (Row total 70) Bus No: Before Yes 40, Before No 30 (Row total 70) Column totals: Before Yes 70, Before No 70, Grand total 140
3. Two groups of students measured plant heights (in cm) after 4 weeks. Both data sets are approximately symmetric with no outliers. Summary statistics: - Data Set A: mean $\bar{x}=22.6$, median $=22.6$, standard deviation $s=1.8$, IQR $=2.4$ - Data Set B: mean $\bar{x}=24.0$, median $=24.0$, standard deviation $s=1.8$, IQR $=2.4$ Which statement best compares the typical value and variability of Data Set A and Data Set B using appropriate measures?
4. A store wants to estimate the population parameter $\mu$, the mean amount (in dollars) customers spend per visit. A random sample of $n=80$ customers had a sample statistic $\bar{x}=24.50$. Students used a simulation with 3,000 repeated random samples of size 80 and recorded the sample mean each time. The simulation showed that about 95% of the simulated sample means were within $$2.00 of $$24.50. Which interval is a reasonable estimate range for the population parameter $\mu$?
5. A researcher records each participant's age (years) and resting heart rate (beats per minute). The scatter plot shows points that form a curved (nonlinear) pattern, and technology reports a correlation coefficient of $r=0.05$. Which statement best interprets the value of $r$ for the relationship between age and resting heart rate?
6. A coach compared two warm-up routines. Using a random number generator, 34 athletes were randomly assigned to Treatment A (dynamic warm-up) and 34 to Treatment B (light jogging). Each athlete then attempted a target drill; success was "hit the target at least 7 times out of 10." In Treatment A, 20 of 34 succeeded ($0.588$). In Treatment B, 19 of 34 succeeded ($0.559$). The observed difference in proportions was $0.588-0.559=0.029$ (A − B). A randomization test shuffled the success/failure outcomes among athletes 2000 times under "no treatment effect." In the 2000 shuffles, 836 produced a difference in proportions of at least $+0.029$ (A − B). Which conclusion is most reasonable about whether Treatment A is more effective than Treatment B?
7. A car rental agency models the total rental cost with $y = 29x + 50,$ where $x$ is the number of days rented and $y$ is the predicted total cost in dollars. What does the y-intercept represent in this context? (Include units.)
8. A science class measured temperatures (in $^\circ$C) of a liquid at 10 different times: 18.2, 18.4, 18.5, 18.5, 18.7, 18.8, 18.8, 19.0, 19.1, 19.3. Which plot best represents the data as a histogram with bin width 0.5: 18.0–18.4, 18.5–18.9, 19.0–19.4?
9. Six runners compete in a race. Gold, silver, and bronze medals are awarded, and order matters because the medal types are different. What is the probability that Ava wins gold and Ben wins silver (with anyone winning bronze)?
10. A large set of reading assessment scores is approximately normal (bell-shaped). Scores have mean $\mu=500$ and standard deviation $\sigma=100$. Approximately what percent of students score between 300 and 700? (Use the empirical rule.)
11. A fair four-sided die with faces $\{1,2,3,4\}$ is rolled twice. All 16 ordered outcomes $(a,b)$ are equally likely. Define the random variable $X=\max(a,b)$ (the larger of the two rolls). Which table correctly represents the theoretical probability distribution of $X$?
12. A random sample of 140 students recorded whether they prefer paper notes or digital notes and whether they feel prepared for quizzes. Using the two-way table, are the events A = "student prefers digital notes" and B = "student feels prepared" independent? Compare $P(A\mid B)$ to $P(A)$.
13. A park manager wants to estimate the population parameter $p$, the proportion of park visitors who arrive by bicycle on weekends. A random sample of $n=100$ visitors had a sample statistic $\hat{p}=0.18$. Students ran a simulation of 10,000 repeated random samples of size 100. They reported: "About 95% of simulated sample proportions were between 0.11 and 0.25." What margin of error is supported by this simulation (in percentage points)?
14. A teacher recorded these quiz scores (out of 20) for one class: 14, 15, 15, 16, 16, 16, 17, 17, 18, 19, 2. Which measure is most affected by the outlier score of 2?
15. Using the outcomes listed in $S=\{\text{red},\text{blue},\text{green},\text{yellow}\}$ for a spinner, event $A$ is landing on a primary color (red, blue, or yellow). Which set represents $A^c$ (the complement of $A$, or not $A$)?
16. According to the sample space $S=\{10,11,12,13,14,15\}$ for selecting one integer, let event $A$ be "the number is a multiple of 3" and event $B$ be "the number is even." Which expression represents $A\cup B$ (A or B)?
17. A warehouse performs a two-step quality check. First, a package is scanned successfully (event $A$). If it is scanned successfully, it is then routed to the correct truck (event $B$). If $P(A)=0.90$ and $P(B\mid A)=0.85$, what is the probability that both $A$ and $B$ occur, $P(A\cap B)$?
18. A teacher compared quiz scores (out of 20 points) from two different sections. Summary statistics: - Data Set A: roughly symmetric, no outliers; mean $= 16.2$, standard deviation $= 1.4$ - Data Set B: roughly symmetric, no outliers; mean $= 15.6$, standard deviation $= 2.3$ Which statement best compares the typical value and variability of Data Set A and Data Set B? Use measures appropriate to the distribution shapes.
19. In a quality-control check at a factory, event $A$ is "a randomly selected item is scratched," and event $B$ is "a randomly selected item has a loose label." Given that $P(A)=0.30$, $P(B)=\tfrac{1}{5}$, and $P(A\cap B)=0.06$, are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$.
20. A fair six-sided die is rolled once. Define the random variable $X$ as the number of even outcomes rolled. Which table correctly represents the probability distribution of $X$?

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