Practice Test 18 - 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 random sample of 120 students at a high school was asked which device they use most for schoolwork (Laptop or Tablet) and whether they participate in an after-school club (Yes or No). Using the two-way table, estimate the probability that a randomly selected student uses a laptop given that the student is in a club. Give your answer as a decimal. Two-way table (counts): - Club Yes & Laptop: 36 - Club Yes & Tablet: 24 - Club No & Laptop: 30 - Club No & Tablet: 30 - Row totals: Club Yes = 60, Club No = 60 - Column totals: Laptop = 66, Tablet = 54 - Grand total: 120
2. A bookstore tracked the number of minutes customers spent in the store on a slow afternoon. The data are: 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 16, 45. Which conclusion about the distribution is best supported?
3. In a school survey, event $A$ is "a randomly selected student plays a school sport" and event $B$ is "a randomly selected student is in the band." The survey results give $P(A)=0.46$, $P(B)=0.38$, and $P(A\cap B)=0.15$. What is $P(A\cup B)$, where "or" is inclusive (sport or band or both)?
4. A streaming service models monthly data usage with $y = 15x + 40,$ where $x$ is the number of hours streamed (hours) and $y$ is the total data used (gigabytes). What does the slope represent in this context? (Include units.)
5. The time it takes a printer to complete a certain job is approximately normal (based on many runs, the distribution is roughly symmetric and bell-shaped). The times have mean $\mu = 30$ seconds and standard deviation $\sigma = 2$ seconds. Approximately what percent of jobs take between 28 and 32 seconds? (Use the empirical rule.)
6. Two machines filled bottles, and the fill amounts (in milliliters) were recorded. Summary statistics: - Data Set A: approximately symmetric; mean $= 502.1$, standard deviation $= 1.1$ - Data Set B: approximately symmetric; mean $= 501.7$, standard deviation $= 0.6$ Which statement best compares the typical value and variability of Data Set A and Data Set B? Use measures appropriate to the distribution shapes.
7. A fair coin is flipped 3 times. The sample space consists of the 8 equally likely sequences (e.g., HHT, TTT). Define the random variable $X$ as the number of heads minus the number of tails in the 3 flips. (For example, HHT gives $X=1$ because $2-1=1$.) Which table correctly represents the theoretical probability distribution of $X$?
8. A card is drawn from a standard 52-card deck first and not replaced. Event $A$ is "the first card is a heart," so $P(A)=\frac{13}{52}$. Event $B$ is "the second card is a king." Given that the first card was a heart, $P(B\mid A)=\frac{4}{51}$. What is the probability that both events occur, $P(A\cap B)$?
9. The number of text messages received in a day by 20 students were: 0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 10, 12, 14, 16, 18. Which statement correctly describes the distribution shown by a histogram of these counts?
10. A game awards tokens. Let $X$ be the number of tokens won in one round. The distribution is: $P(X=0)=\tfrac{1}{6}$, $P(X=3)=\tfrac{1}{2}$, $P(X=6)=\tfrac{1}{3}$. What is the expected value of $X$ (in tokens)? (Note: the expected value may not be a possible outcome.)
11. A store is deciding between two rebate promotions. For each customer purchase, exactly one rebate amount is awarded according to the listed probabilities. Strategy A (mail-in rebate): - $0 with probability 0.40 - $10 with probability 0.50 - $50 with probability 0.10 Strategy B (instant rebate): - $6 with probability 1.00 Which option would result in a higher average rebate per purchase over many purchases?
12. A jar contains 2 green balls and 2 yellow balls. Two balls are drawn without replacement. Let $X$ be the number of green balls drawn. Which table correctly represents the probability distribution of $X$?
13. A student used the model $\hat{y}=100-5x$ to predict the number of pages (y) remaining in a book after x days. On day 6, the model predicted 70 pages remaining, but the actual number of pages remaining was 74. For x = 6, what does the residual mean in context?
14. A fair six-sided die is rolled once (outcomes $1$ through $6$ are equally likely). Define the random variable $X$ as $X=1$ if the roll is prime ($2,3,5$) and $X=0$ otherwise ($1,4,6$). What is the expected value of $X$ based on this theoretical model?
15. According to $S=\{\text{Mon},\text{Tue},\text{Wed},\text{Thu},\text{Fri},\text{Sat},\text{Sun}\}$, one day is chosen at random. Event $A=\{x\in S\mid x\text{ is a weekday}\}$. Which set of outcomes represents event $A$?
16. A cafeteria survey defines $X$ as the number of days per week a student buys lunch at school. The distribution is: - $P(X=0)=\tfrac{1}{5}$ - $P(X=2)=\tfrac{2}{5}$ - $P(X=5)=\tfrac{2}{5}$ What is the expected value of $X$ (in days per week)? (Note: the expected value may not be a possible outcome.)
17. School context: Two events are defined. Event A: a student participated in class discussion today. Event B: a student drank coffee before school today. A student claims, "Since the chance of participating is higher among coffee drinkers, coffee makes students participate more." Which statement best describes what "the chance of A given that B" means in this situation?
18. A factory tracks two types of minor defects. Event $A$ is "a randomly selected item has a scratch," and event $B$ is "a randomly selected item has a dent." The rates are $P(A)=0.18$, $P(B)=0.14$, and $P(A\cap B)=0.06$. What is the probability that an item has a scratch or a dent (inclusive)?
19. A school district report states: "Screen time is linked to lower reading scores." The report combines data from 5 schools (total $n=1{,}200$ students) and shows a negative association overall. However, within each school, the association is weak or slightly positive. Which critique best evaluates the claim?
20. A teacher posted the quiz scores (out of 20) for one class: 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 16, 19, 20. Which measure of spread is most affected if the score 20 is replaced with a 2 because one student was absent and received a zero that was entered incorrectly as 2?

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