Practice Test 1 - Statistics and Probability

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

A school website headline reads: "New online homework system improves grades." The school piloted the system by letting students choose whether to use it. At the end of the quarter, the average grade in math was 88 for students who chose the system (n=70) and 82 for students who did not (n=90). Which critique best evaluates the claim?
A school nurse recorded the number of students visiting the office each day for 15 days: 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 13, 14, 28 Which measure is most affected by the outlier (28 visits)?
A streaming service is choosing a recommendation strategy for a new show. Based on prior launches: - Strategy Broad (recommend to many users): 12% of recommended users watch at least one full episode. - Strategy Targeted (recommend to fewer, high-likelihood users): 22% of recommended users watch at least one full episode. But there is uncertainty about whether the show has "wide appeal" or "niche appeal": - There is a 40% chance it has wide appeal, in which case Broad yields 15% watch-through and Targeted yields 18%. - There is a 60% chance it has niche appeal, in which case Broad yields 10% watch-through and Targeted yields 25%. The service's goal is to maximize the probability that a randomly selected recommended user actually watches a full episode (i.e., higher conversion rate, not total viewers). Which strategy is most reasonable given the probabilities?
A website user session proceeds in two steps. A user first clicks an ad (event $A$ ). If the user clicks the ad, the user then makes a purchase (event $B$ ). If $P(A)=0.20$ and $P(B\mid A)=0.35$ , what is the probability that both $A$ and $B$ occur (i.e., $P(A\cap B)$ )?
A fitness app company randomly assigned 200 volunteers to either walk 2,000 extra steps per day or to keep their usual step count for 6 weeks. At the end, the group assigned to walk extra steps had a lower average resting heart rate. Can we conclude that increasing daily steps causes a change in resting heart rate? Why or why not?
A researcher wanted to test whether background music affects reading speed. They randomly assigned 90 students to read the same passage either in silence or with instrumental music playing, then measured reading speed (words per minute). The music group read more slowly on average. Which statement best describes what the data show?
According to $S=\{(H,H),(H,T),(T,H),(T,T)\}$ for a random experiment where two coins are flipped (first coin, second coin), event $A$ is "exactly one head" and event $B$ is "the first coin is heads." Which set of outcomes represents event $A$ ?
A survey asks customers about preferences. Event $A$ is "a customer prefers online shopping," event $B$ is "a customer uses coupons." The survey reports $P(A)=0.65$ , $P(B)=0.40$ , and $P(A\cap B)=0.28$ . What is the probability that $A$ or $B$ occurs (inclusive "or")?
A teacher listed the time (in minutes) it took 13 students to finish a logic puzzle: 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 14, 28 How does the outlier 28 minutes affect the mean and median finish times?
A spinner has 4 equal sections labeled 1, 2, 3, and 4. The spinner is spun once. Define the random variable $X$ as the value shown. Which table correctly represents the probability distribution of $X$ ?
A machine fills cereal boxes, and the fill weights are approximately normal. The mean fill weight is $\mu=500$ g with standard deviation $\sigma=10$ g. Approximately what percent of boxes have fill weights between 490 g and 510 g? (Use the empirical rule.)
Times (in minutes) to complete a certain assembly task are roughly symmetric and bell-shaped, so a normal model is appropriate. Completion times are modeled as $N(\mu=40,\,\sigma=4)$ . Approximately what percent of tasks take more than $48$ minutes? (Use the empirical rule.)
In a randomized experiment at a community center, 80 participants were randomly assigned using a random number generator to either Treatment A (a new puzzle game) or Treatment B (a classic puzzle game), 40 in each group. After 10 minutes, success was defined as "solved the puzzle." In Treatment A, 26 of 40 succeeded ( $0.65$ ). In Treatment B, 18 of 40 succeeded ( $0.45$ ). The observed difference in proportions was $0.65-0.45=0.20$ (A − B). A randomization test shuffled the success/failure outcomes among the 80 participants 5000 times under "no treatment effect." In the 5000 shuffles, 41 produced a difference in proportions of at least $+0.20$ (A − B). Do the results provide evidence that Treatment A is more effective than Treatment B?
A gym chain wants to estimate the mean number of visits per month among all members of its Downtown location. It randomly selects 60 member accounts from the Downtown membership list and calculates the sample mean visits last month as 6.2. What is a reasonable inference based on the sample?
A coach randomly assigned 60 runners to follow one of two training plans for 8 weeks: Plan A required 3 practice sessions per week, and Plan B required 5 practice sessions per week. At the end, the group assigned to 5 sessions per week had faster average 5K times (lower times) than the group assigned to 3 sessions per week. Can we conclude that practicing more causes faster 5K times? Why or why not?
School context: At a high school, event A is "a student is in the drama club," and event B is "a student is in the chess club." A counselor asks, "Among students who are in the chess club, about how often are they also in the drama club?" Which statement best describes what "the chance of A given B" means in this situation?
A weather station models the temperature during a morning warm-up with $y = -4 + 1.5x,$ where $x$ is the number of hours after 6:00 a.m. (hours) and $y$ is the temperature (degrees Celsius). Is the y-intercept meaningful in this context? Choose the best statement.
A fair coin is flipped twice. The sample space is $\{HH, HT, TH, TT\}$ . Define the random variable $X$ as the number of heads obtained. Which table correctly represents the probability distribution of $X$ ?
A website tracked the number of pages visited by a user during 100 sessions. The observed counts were: 1 page (25 sessions), 2 pages (40 sessions), 3 pages (20 sessions), 4 pages (15 sessions). Let $X$ be the number of pages visited in a randomly selected session. Using the empirical distribution from the observed relative frequencies (so probabilities are estimates), what is the expected value of $X$ based on the empirical distribution?
A school wants to estimate the population parameter $\mu$ , the mean number of minutes students spend on homework per night. A random sample of $n=40$ students reported a sample statistic (sample mean) of $\bar{x}=78$ minutes. Students ran a simulation by repeatedly taking 2,000 random samples of size 40 from a large list meant to represent the school and recording each simulated sample mean. They found that about 95% of the simulated sample means were within 6 minutes of 78. What margin of error is supported by the simulation for estimating $\mu$ ?

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