Practice Test 37 - 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 park department wants to estimate the proportion of all households in the city that visited a city park at least once in the past month. It randomly selects 150 households from the city's address list and finds that 72 report visiting a park. Which statement correctly interprets what can and cannot be inferred from this sample?
2. A city models electricity use ($y$) from average daily temperature ($x$) using a linear model $\hat{y}=90-2x$. The residuals ($\text{residual}=y-\hat{y}$) for selected temperatures are shown. Which statement best describes how well the model fits the data based on the residuals?
3. A prize wheel costs $1 per spin. Outcomes: win $7 with probability $\tfrac{1}{10}$, win $1 with probability $\tfrac{3}{10}$, win $0 with probability $\tfrac{6}{10}$. From the player's perspective, what is the expected value (long-run average net gain/loss) of 1 spin?
4. A student council post claims: "Most students want longer lunch." Their poll asked 120 students standing in line at the cafeteria during lunch on Tuesday; 78 said they want longer lunch. The post concludes: "Therefore, most students at the school want longer lunch." Which critique best evaluates the claim?
5. In a group of 50 employees, 20 work remotely (event $B$) and 8 both work remotely and are on the morning shift (event $A\cap B$), where event $A$ is "an employee is on the morning shift." Based on the information given, what is $P(A\mid B)$? Give your answer as a simplified fraction.
6. A student randomly selects one of five index cards labeled A, B, C, D, E (each equally likely). Define the random variable $X$ by assigning $X=0$ if the letter is a vowel and $X=1$ if the letter is a consonant. Which table correctly represents the probability distribution of $X$?​
7. A bike shop wants to estimate the mean repair cost for all repairs completed last month. The owner randomly selects 30 repair invoices from all invoices dated last month and computes the sample mean repair cost. Why might the owner get a different sample mean if a different random sample of 30 invoices were selected?
8. A deck of cards is modified so that it contains 40 cards total. Event $A$ is "the card drawn is a heart," and event $B$ is "the card drawn is a face card." In this deck, $P(B)=\frac{1}{5}$ and $P(A\cap B)=\frac{1}{20}$. Based on the information given, what is $P(A\mid B)$? Give your answer as a fraction in simplest form.
9. A movie theater models total ticket revenue for a showing with $y = 200 + 9x$, where $x$ is the number of tickets sold (tickets) and $y$ is the total revenue (dollars). What does the slope represent in this context? Include units in your interpretation.
10. A quality-control check tracks two events on a randomly selected item: Event $A$ = "the item passes the visual inspection," and event $B$ = "the item passes the weight test." Given that $P(A)=0.6$, $P(B)=\tfrac{1}{2}$, and $P(A\cap B)=0.25$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
11. A batch of resistors has resistances that are approximately normal (roughly symmetric and bell-shaped). Resistances are modeled as $N(\mu=200\ \Omega,\,\sigma=5\ \Omega)$. Approximately what percent of resistors have resistance below $195\ \Omega$? (Use the empirical rule.)
12. A machine produces a part and then a sensor checks it. First, the part must be within size tolerance (event $A$). Then, given it is within tolerance, the sensor must correctly label it as acceptable (event $B$). Suppose $P(A)=0.92$ and $P(B\mid A)=0.95$. What is the probability that both $A$ and $B$ occur?
13. A 4-digit code is formed by arranging 4 different digits chosen from 1 through 6, with no repetition. Order matters. What is the probability that the code starts with 1?
14. School context: Two events are defined. Event A: a student gets an A on the English essay. Event B: a student met with the writing center tutor. The department chair says, "The overall A rate is 25%, but among students who met with the tutor, the A rate is 35%." Which statement best describes what "the chance of A given that B" means in this situation?
15. A market survey defines event $A$ as "a randomly selected customer has bought brand X in the last month," and event $B$ as "a randomly selected customer has used a coupon in the last month." Suppose $P(A)=0.33$, $P(B)=0.41$, and $P(A\cap B)=0.12$. Using inclusive or, what is $P(A\cup B)$?
16. Two cards are drawn with replacement from a standard 52-card deck. Let $A$ be "the first card is an ace" and $B$ be "the second card is an ace." Given that $P(A)=\tfrac{1}{13}$, $P(B)=\tfrac{1}{13}$, and $P(A\cap B)=0.006$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
17. A principal announces: "The new hallway rule reduced tardiness." The school compared two groups of randomly selected students. Group 1 followed the new rule for one month; Group 2 followed the old rule for one month. Students were randomly assigned to groups. Group 1 averaged 3 tardies per student, and Group 2 averaged 5 tardies per student. Which statement best describes whether the conclusion is justified?
18. A company finds $P(A)=0.50$, $P(B)=0.40$, and $P(A\cap B)=0.30$, where event $A$ is "an employee completed training" and event $B$ is "an employee received a promotion." Based on the information given, are $A$ and $B$ independent? Use conditional probability by comparing $P(A\mid B)$ and $P(A)$.
19. A school compared the number of absences in a semester (x) and the student's semester GPA on a 4.0 scale (y) for a random sample of students. The correlation coefficient is $r=-0.74$. Which statement best interprets the value of $r$?
20. Two delivery routes were timed (in minutes) over several days. Summary statistics: - Data Set A: right-skewed with a clear high outlier; median $= 32$, IQR $= 6$ - Data Set B: right-skewed with a clear high outlier; median $= 29$, IQR $= 10$ Which statement best compares the typical value and variability of Data Set A and Data Set B? Use measures appropriate to the distribution shapes.

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