Practice Test 4 - 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 teacher wants to know whether background music affects quiz performance. The teacher randomly assigns half the class to take a 20-question quiz with quiet background music and the other half to take the same quiz in silence. The music group has a higher average score. Which statement is the most reasonable conclusion?
2. Seven players want to be team captain: 1, 2, 3, 4, 5, 6, 7. A method is fair if each player has probability $\tfrac17$ of being chosen. Which procedure is fair?
3. A charity is choosing between two fundraising phone scripts. The charity wants the higher expected net donation per call over many calls. (Net donation = donation received minus the cost of the call.) Strategy A: Each call costs $1. With probability 0.15, the person donates $20; with probability 0.85, the person donates $0. Strategy B: Each call costs $2. With probability 0.10, the person donates $35; with probability 0.90, the person donates $0. Which strategy has the greater expected value of net donation over many repetitions?
4. A card is drawn from a standard 52-card deck without replacement, and then a second card is drawn. Let event $A$ be "the first card is a heart," so $P(A)=\frac{13}{52}$. Let event $B$ be "the second card is a heart given the first card was a heart," so $P(B\mid A)=\frac{12}{51}$. What is the probability that both cards drawn are hearts (i.e., $P(A\cap B)$)?
5. A news article reports: "Cities with higher average daily temperature tend to have higher average daily electricity use" (a positive association), based on utility records from many cities. The researchers did not change temperatures; they only compared existing city data. Which statement best describes what the data show?
6. A math department wants to know whether allowing students to choose their own seats affects quiz performance. In one class period, students choose their seats as usual; in another class period, the teacher assigns seats alphabetically. After two weeks, both classes take the same quiz, and the teacher compares average scores. Students were not randomly assigned to class periods. What can the teacher reasonably conclude about causation and generalization?
7. A die is claimed to be fair, so $P(6)=\tfrac{1}{6}$. You roll it 30 times and get six 6s. To evaluate this, you simulate 120 runs of 30 fair-die rolls and count how often you get 6 or more sixes. In the simulations, 41 out of 120 runs had 6 or more sixes. Would this result cause you to question the fair-die model? Why?
8. A delivery service models the predicted delivery time with $y = -1.5x + 40,$ where $x$ is the number of delivery drivers working (drivers) and $y$ is the predicted average delivery time (minutes). What does the slope represent in this context (include units)?
9. Two vending machines were monitored for the number of items sold per day. Both data sets are approximately symmetric with no outliers. Summary statistics: - Data Set A: mean $\bar{x}=52.0$, median $=52.0$, standard deviation $s=4.0$, IQR $=6$ - Data Set B: mean $\bar{x}=49.5$, median $=49.5$, standard deviation $s=2.0$, IQR $=3$ Which statement best compares the typical value and variability of Data Set A and Data Set B using appropriate measures?
10. A student randomly selects one card from a set of 5 cards labeled A, B, C, D, and E (each card equally likely). Define the random variable $X$ as follows: $X=0$ if A or B is selected, $X=1$ if C is selected, and $X=2$ if D or E is selected. Which table correctly represents the probability distribution of $X$?
11. School context: Two events are defined. Event A: a student got to school before the first bell today. Event B: a student rode the bus today. A principal says, "The chance of being early given that you rode the bus is 55%." Which statement best describes what "the chance of A given that B" means in this situation?
12. Adult male heights in a certain region are approximately symmetric and bell-shaped, so a normal model is appropriate. Assume heights are normally distributed with mean $\mu = 70\text{ in}$ and standard deviation $\sigma = 2\text{ in}$. Approximately what percent of adult males are between 68 inches and 72 inches tall?
13. A meteorologist modeled the relationship between humidity $x$ (%) and a heat index measure $y$ using the linear model $\hat{y}=20+0.6x$. The residual plot below shows points scattered around 0 with no obvious pattern. Which statement best describes how well the model fits the data?
14. A student measured the number of pages read each day for 14 days: 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 18, 19, 20. Which plot best represents the data as a dot plot on a number line?
15. A city is piloting two different sensors to detect when a parking space becomes available. Sensor X: - When a space is actually available, it reports "available" 88% of the time. - When a space is actually occupied, it still reports "available" 12% of the time. Sensor Y: - When a space is actually available, it reports "available" 78% of the time. - When a space is actually occupied, it still reports "available" 4% of the time. On a busy street, only about 15% of the time a given space is actually available (base rate 15%). The city's goal is to reduce drivers circling based on false "available" signals, even if that means missing some real openings. Which strategy is most reasonable given the probabilities?
16. A die is rolled once. Let $A$ be "the result is even" and $B$ be "the result is greater than 4." Given that $P(A)=\tfrac{1}{2}$, $P(B)=\tfrac{1}{3}$, and $P(A\cap B)=0.20$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
17. A researcher fits the line $y = 1.8x - 4$ to predict a runner's heart rate. Here, $x$ is running speed (miles per hour) and $y$ is heart rate (beats per minute). What does the slope represent in this context? Include units.
18. Two fair six-sided dice are rolled (one red, one blue), so the $36$ ordered outcomes $(r,b)$ are equally likely. Define the random variable $X$ as the sum $X=r+b$. The outcomes map to $X$ by sums (e.g., $\{(1,1)\}\to 2$, $\{(1,2),(2,1)\}\to 3$, ..., $\{(6,6)\}\to 12$). Which table correctly represents the theoretical probability distribution of $X$?
19. Two classes, Class 1 and Class 2, both want to use the gym at the same time. The principal wants a fair method, meaning each class should have probability $1/2$ of getting the gym today. Which procedure is fair?
20. A coin is flipped and then a number cube (1–6) is rolled. All 12 outcomes are equally likely: $\{H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6\}$ Event $B$: the coin shows heads. Event $A$: the number rolled is even. Using the outcomes listed, what is $P(A\mid B)$? Write your answer as a simplified fraction.

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