Practice Test 12 - 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 school wants to find out what fraction of students prefer digital textbooks over printed textbooks. Students are invited to answer an online poll posted on the school's social media page, and 350 students respond. Which statement about the results is most accurate?
2. A school wants to estimate the population parameter $p$, the proportion of all students at the school who usually bring a reusable water bottle. A random sample of $n=200$ students is surveyed, and the sample statistic is $\hat{p}=0.58$ (58%). Students run a simulation of random sampling by repeatedly taking 1000 random samples of size 200 from a model where the true proportion is unknown, and they record the simulated sample proportions. They find that about 95% of the simulated $\hat{p}$ values are within 0.04 (4 percentage points) of 0.58. Which interval is a reasonable estimate range for the population parameter $p$? Write your answer as a percent interval.​
3. A student recorded the number of text messages received each day for 13 days: 5, 6, 6, 7, 7, 7, 8, 9, 10, 10, 11, 12, 14. Which statement correctly describes the distribution in a dot plot?
4. A factory tracks two conditions on items: event $A$ is "an item is underweight," and event $B$ is "an item is overfilled with packaging." Given that $P(A)=0.15$, $P(B)=\tfrac{2}{5}$, and $P(A\cap B)=0.05$, are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$.
5. A cafeteria manager claims: "Most students prefer the new pizza recipe." To support this, the manager asked the first 40 students who entered the cafeteria on Monday and found 28 said they preferred the new recipe. Which statement best describes whether the conclusion is justified?
6. A cafeteria tracked how many items students bought at lunch. Data Set A (items per student): 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4 Data Set B (items per student): 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 7 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 researcher models the amount of fertilizer applied ($x$) and crop yield ($y$) with a linear model $\hat{y}=40+4x$. A residual plot for this model is shown. Which statement best describes how well the model fits the data?
8. A random sample of 100 students was asked whether they ride the bus to school and whether they are in band. Using the two-way table, estimate the probability that a randomly selected student is in band given that the student rides the bus. Give your answer as a fraction in simplest form.
9. A student is selected at random from a large group. Event $A$ is "the student is in Grade 11," and event $B$ is "the student is enrolled in an elective art class." Given that $P(A)=\tfrac{2}{5}$, $P(B)=0.30$, and $P(A\cap B)=0.12$, are $A$ and $B$ independent? Justify using probabilities (compare $P(A\cap B)$ to $P(A)P(B)$).
10. A product review site tracks two labels. Event $A$ is "a randomly selected review mentions 'easy to use'," and event $B$ is "a randomly selected review mentions 'good value'." The site reports $P(A)=0.33$, $P(B)=0.29$, and $P(A\cap B)=0.11$. What is the probability that a review mentions 'easy to use' or 'good value' (inclusive)?
11. Based on the sample space shown, $S=\{\text{pencil},\text{pen},\text{marker},\text{crayon},\text{chalk}\}$, one item is selected at random. Event $A=\{\text{pen},\text{marker}\}$. Event $B$ is the event that the item contains the letter "c" in its name. Which expression represents $A\cup B$ (A or B)?
12. A student applies for a scholarship in two stages. First, the student must submit the application on time (event $A$). The probability of submitting on time is $P(A)=0.70$. Then, given the application was submitted on time, the probability it is approved is $P(B\mid A)=0.30$. What is the probability that both events occur, $P(A\cap B)$?
13. A band director recorded the 20 practice times (in minutes) for students over a week: 5, 10, 10, 15, 15, 15, 20, 20, 20, 20, 25, 25, 30, 30, 35, 40, 45, 50, 55, 60. Which statement correctly describes the distribution shown by a histogram of these times?
14. Based on the sample space $S=\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)\}$ for rolling a 2-sided die (values 1–2) and a 3-sided die (values 1–3) and recording (first, second), let event $A$ be "the sum is 4" and event $B$ be "the first number is 2." Which set represents $A\cup B$ (A or B)?
15. A school models the number of books checked out ($y$) from the number of library visits ($x$) using $\hat{y}=1.2x+5$. The residuals ($\text{residual}=y-\hat{y}$) are shown. Which statement best describes how well the model fits the data based on the residuals?
16. A researcher models the relationship between study time $x$ (hours) and test score $y$ (points) with the linear model $\hat{y}=50+5x$. A residual plot of $\text{residual}=y-\hat{y}$ versus $x$ shows residuals that are mostly negative for small $x$, positive for middle $x$, and negative again for large $x$, forming a clear curved pattern around the horizontal 0 line. Which statement best describes how well the model fits the data?
17. A student surveyed 150 classmates and found that those who reported spending more dollars per week on coffee tended to report fewer hours of sleep per night (a negative association). No one was assigned to drink coffee; the student only collected survey responses. Can we conclude that spending more on coffee causes less sleep? Why or why not?
18. A fair die with faces 1–6 is rolled once. Outcomes: {1, 2, 3, 4, 5, 6} Let event $A$ = "the roll is a multiple of 3" and event $B$ = "the roll is even." Based on the outcomes listed, what is $P(A\mid B)$? Give your answer as a fraction.
19. A student randomly selects one of the 15 equally likely stickers listed below: Cat-Small, Cat-Small, Cat-Large, Dog-Small, Dog-Large, Dog-Large, Fish-Small, Fish-Small, Fish-Large, Bird-Small, Bird-Large, Bird-Large, Bird-Large, Turtle-Small, Turtle-Large Let event $A$ be "the sticker is Large," and let event $B$ be "the sticker is Bird." Using the outcomes listed, what is $P(A\mid B)$? Give your answer as a fraction.
20. A consumer analyst recorded the age of a used car in years (x) and its resale price in dollars (y) for several cars of the same model. The correlation coefficient is $r=-0.91$. Which statement best interprets the value of $r$?

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