Practice Test 19 - Statistics and Probability

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

A city's monthly report shows that when average outdoor temperature (quantitative) is higher, electricity use per household (quantitative) is also higher, indicating a positive correlation. The report uses historical meter readings and weather records; no variables were controlled or assigned (observational data). Which statement is the most reasonable conclusion?
A coach compared sprint times (in seconds) for two training groups. Both distributions are approximately symmetric with no outliers. Summary statistics: - Data Set A: mean $\bar{x}=12.4$ , median $=12.4$ , standard deviation $s=0.6$ , IQR $=0.8$ - Data Set B: mean $\bar{x}=12.0$ , median $=12.0$ , standard deviation $s=1.1$ , IQR $=1.5$ Which statement best compares the typical value and variability of Data Set A and Data Set B using appropriate measures?
A jar contains 9 equally likely tokens: Numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9 Let event $A$ be "the number is prime" and event $B$ be "the number is less than 6." What fraction of outcomes in $B$ also belong to $A$ ? Give your answer as a simplified fraction.
Two brands of light bulbs were tested for lifetime (in hours). Data Set A (Brand A): 980, 995, 1005, 1010, 1015, 1020, 1025, 1030, 1040, 1055 Data Set B (Brand B): 900, 960, 990, 1000, 1005, 1010, 1015, 1020, 1030, 1100 Which statement best compares the typical value and variability of Data Set A and Data Set B? Use measures appropriate to the distribution shapes.
A game uses a bag containing 5 equally likely tickets labeled $A,B,C,D,E$ . One ticket is drawn at random. Define the random variable $X$ by the mapping: $X=-2$ if $A$ is drawn; $X=0$ if $B$ or $C$ is drawn; and $X=3$ if $D$ or $E$ is drawn. What is the expected value of $X$ based on this theoretical model?
A school randomly assigned 25 students to Treatment A (music while working) and 25 students to Treatment B (silence) by drawing names from a hat and alternating assignments. Each student then completed the same logic worksheet, and the outcome was the time (in minutes) to finish. The mean time for Treatment A was 18.2 minutes and for Treatment B was 19.0 minutes, so the observed difference in means was $18.2-19.0=-0.8$ minutes (A − B). A randomization test shuffled the treatment labels 3000 times under "no treatment effect." In 3000 shuffles, 27 produced a difference in means of at most $-0.8$ minutes (A − B). Which conclusion is most reasonable about the treatment effect (A − B), where more negative values favor Treatment A (faster times)?
Three songs—Song A, Song B, and Song C—are finalists for the school dance. The DJ wants a fair tie-breaker to decide which song plays first, meaning each song must have probability $1/3$ of being selected. Which procedure is fair?
A quality-control inspector checks 80 items and records the number of defects on each item. The observed counts are: 0 defects (50 items), 1 defect (20 items), 2 defects (8 items), 3 defects (2 items). Let $X$ be the number of defects on a randomly selected item. Using the relative frequencies from this sample as estimated probabilities (they may vary from sample to sample), what is the expected value of $X$ based on the empirical distribution?
A factory monitors the thickness of a coating, which is approximately normal (the histogram is roughly symmetric and bell-shaped). Thickness has mean $\mu = 1.20$ mm and standard deviation $\sigma = 0.05$ mm. Approximately what percent of items have thickness below 1.15 mm? (Use the empirical rule.)
Adult male heights in a large population are roughly normal (approximately symmetric and bell-shaped). Suppose heights have mean $\mu=70$ in and standard deviation $\sigma=3$ in. Approximately what percent of men are below 64 inches? (Use the empirical rule.)
At a school, event $A$ is that a randomly selected student takes Algebra, and event $B$ is that the student takes Biology. The probabilities are $P(A)=0.55$ , $P(B)=0.46$ , and $P(A\cap B)=0.30$ . What is the probability that $A$ or $B$ occurs (inclusive or: Algebra, Biology, or both)?
A website claims that 60% of visitors click an ad, so $P(\text{click})=0.60$ . In a sample of 30 visitors, only 9 clicked. To evaluate this, you simulate many samples of 30 visitors using $P(\text{click})=0.60$ and look for results 9 or fewer clicks. In 100 simulated samples, 2 had 9 or fewer clicks. What does the simulation suggest about how unusual the observed result is?
A class tested two note-taking methods. The teacher randomly assigned 28 students to Treatment A (guided notes) and 28 students to Treatment B (blank notes). After a lesson, students took the same 12-question quiz. Treatment A's mean score was 9.8 and Treatment B's mean score was 9.1, so the observed difference in means was $9.8-9.1=0.7$ (A − B). A randomization test shuffled the A/B labels 1500 times under "no treatment effect." In the 1500 shuffles, 735 produced a difference in means of at least $+0.7$ (A − B). Do the results provide evidence that Treatment A is more effective than Treatment B?
A quality-control team inspects items from a production line. Let event $A$ be "a randomly selected item is scratched" and event $B$ be "a randomly selected item has a loose label." Given that $P(A)=0.20$ , $P(B)=\tfrac{3}{10}$ , and $P(A\cap B)=0.06$ , are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$ .
In a class, event $A$ is "a randomly selected student is taking Algebra," and event $B$ is "a randomly selected student is taking Biology." Suppose $P(A)=0.60$ , $P(B)=0.45$ , and $P(A\cap B)=0.30$ . What is the probability that $A$ or $B$ occurs (inclusive "or")?
A player is choosing between two carnival games. Compute expected net winnings per play (prize value minus entry fee) to decide which is better over many plays. Strategy A (entry fee $5): - 0.75 probability of winning a $0 prize - 0.20 probability of winning a $10 prize - 0.05 probability of winning a $50 prize Strategy B (entry fee $3): - 0.60 probability of winning a $0 prize - 0.35 probability of winning a $6 prize - 0.05 probability of winning a $30 prize Which strategy has the greater expected value (average net winnings) over many repetitions?
A park district wants to estimate the population parameter $p$ , the proportion of all residents who visited a park in the last month. A random sample of $n=150$ residents found a sample statistic $\hat{p}=0.48$ (48%). Students used simulation/repeated sampling and reported: in 1,000 simulated samples of size 150, about 95% of the simulated sample proportions were within 0.05 of 0.48. Which interval is a reasonable estimate range for the population parameter $p$ ? (Give the interval in decimals.)
A company tracked the number of customer support tickets handled per day by two employees over 14 days. Data Set A (Employee A): 18, 19, 20, 20, 21, 21, 22, 22, 22, 23, 23, 24, 24, 25 Data Set B (Employee B): 15, 16, 18, 19, 20, 20, 21, 21, 22, 23, 24, 26, 27, 29 Based on the data, which statement best compares the typical number of tickets and variability using appropriate measures?
A gym wants to estimate the population parameter $\mu$ , the mean number of times per week all members work out. A random sample of $n=25$ members gave a sample statistic $\bar{x}=3.6$ workouts/week. Students simulated 2,000 repeated random samples of size 25 and recorded each sample mean. They found that about 95% of simulated sample means were between 3.0 and 4.2. Which interval is a reasonable estimate range for the population parameter $\mu$ ?
A library wants to estimate the population parameter $p$ , the proportion of all patrons who prefer e-books over print books. A random sample of 120 patrons found that 45 preferred e-books, so the sample statistic is $\hat p = 45/120 = 0.375$ (37.5%). Students ran a simulation of 2,000 repeated random samples of size 120, and the middle 95% of simulated sample proportions ran from 0.29 to 0.46. What margin of error is supported by the simulation for estimating the population parameter $p$ ? (Give the margin of error as a proportion.)

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