Practice Test 45 - 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 die is rolled once. Let event $A$ be "the roll is greater than 4" and event $B$ be "the roll is even." Given that $P(A)=\tfrac{1}{3}$, $P(B)=0.50$, and $P(A\cap B)=\tfrac{1}{6}$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
2. A researcher uses $\hat{y}=3x+5$ to predict the number of pages read (y) from the number of days in a reading program (x). Residuals are defined as $\text{residual}=y-\hat{y}$. For $x=10$ days, the residual is $+6$. What does this residual mean in context?
3. At a school, two events are defined: Event A = "a student is in the school band," and Event B = "a student is in the honors math class." The principal says, "Of those who are in honors math, 30% are also in band." Which statement best describes what "the chance of A given that B" means in this situation?
4. A linear model is used to predict a runner's heart rate ($y$, beats per minute) from running speed ($x$, mph): $\hat{y}=60+8x$. Residuals ($y-\hat{y}$) for several speeds are shown. Which statement best describes how well the model fits the data?
5. A coach recorded whether students play a fall sport and whether they prefer morning or afternoon practice. Use the two-way table to answer the question. Which statement best describes the association between playing a fall sport and preferring morning practice?

   	Prefer Morning	Prefer Afternoon	Total
   Plays Fall Sport	44	36	80
   Does Not Play	30	70	100
   Total	74	106	180

6. A fair die is rolled once. The sample space is $\{1,2,3,4,5,6\}$. Define the random variable $X$ as the number of dots showing if the roll is 1, 2, or 3, and define $X=3$ if the roll is 4, 5, or 6. (So multiple outcomes can map to the same $X$ value.) Which table correctly represents the probability distribution of $X$?
7. According to the sample space $S=\{\text{apple},\text{banana},\text{carrot},\text{dates}\}$ for randomly selecting one item, event $A$ is "the item is a fruit" and event $B$ is "the item starts with the letter b." Which set represents $A\cap B$ (A and B)?
8. A city library wants to estimate the proportion of all registered library members who prefer e-books over printed books. The library has a list of all 8,000 members and uses a random number generator to select 120 members to survey. In the sample, 48 say they prefer e-books. Which statement is a reasonable inference based on the sample?
9. Two-step experiment with replacement: A jar contains many tokens. A token is drawn, recorded, and replaced; then a second token is drawn. Event $A$ is "the first token is green," and event $B$ is "the second token is green." Given that $P(A)=0.30$, $P(B)=0.30$, and $P(A\cap B)=0.09$, are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$.
10. A jar contains 12 equally likely marbles labeled: R1, R2, R3, R4, R5, R6, B1, B2, B3, B4, B5, B6. Event $B$: the marble is blue (B). Event $A$: the number is a multiple of 3. Using the labels, what is $P(A\mid B)$? Give your answer as a simplified fraction.
11. A two-stage experiment is performed: first choose a shape from $\{\text{Circle},\text{Square},\text{Triangle}\}$, then choose a color from $\{\text{Red},\text{Blue}\}$. All 6 outcomes are equally likely: $\{CR, CB, SR, SB, TR, TB\}$ (For example, $SB$ means Square-Blue.) Event $B$: the outcome is Blue. Event $A$: the shape is a Circle or Triangle. What fraction of outcomes in $B$ also belong to $A$? Give your answer as a simplified fraction.
12. A clinic uses a screening test for a condition in a low-risk population. In this population, about $3\%$ of people actually have the condition. The test has: - Sensitivity $80\%$ - Specificity $90\%$ A patient tests positive. The clinic must decide what to do next. Options: A) Start treatment immediately. B) Order a more accurate confirmatory test before treatment. C) Ignore the result because the patient is low-risk. The clinic's goal is to minimize unnecessary treatment while still taking positives seriously. Which strategy is most reasonable given the probabilities?
13. A fair six-sided die is rolled once (outcomes $1$–$6$ equally likely). Define the random variable $X$ by the mapping: $X=0$ if the outcome is odd ($1,3,5$) and $X=2$ if the outcome is even ($2,4,6$). What is the expected value of $X$ based on this theoretical model?
14. A recreation center wants to estimate the population parameter $\mu$, the mean number of days per week adults in the community exercise. A random sample of $n=60$ adults reported a sample statistic $\bar{x}=3.9$ days. Using a simulation of 1,000 repeated random samples of size 60, students found that the middle 95% of simulated sample means ran from 3.4 to 4.4 days. Which interval is a reasonable estimate range for the population parameter $\mu$?
15. A student rolls a fair six-sided die once. Event $A$ is "the roll is greater than 4," and event $B$ is "the roll is even." Given that $P(A)=\tfrac{1}{3}$, $P(B)=\tfrac{1}{2}$, and $P(A\cap B)=\tfrac{1}{6}$, are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$.
16. A study fits the line $y = 1.8x + 62,$ where $x$ is the number of minutes a student studies for a quiz (minutes) and $y$ is the predicted quiz score (points). What does the y-intercept represent in this context (include units)?
17. Two teachers recorded the number of minutes it took students to finish a puzzle. Data Set A is roughly symmetric with no outliers; Data Set B is right-skewed with one unusually long time. Summary statistics: - Data Set A: mean $\bar{x}=18.0$, median $=18.0$, standard deviation $s=2.1$, IQR $=3$ - Data Set B: mean $\bar{x}=20.6$, median $=19.0$, standard deviation $s=5.8$, IQR $=4$ From the summary statistics, which statement best compares the typical value and variability of Data Set A and Data Set B using appropriate measures?
18. A student pays $3 to play a game once. A bag contains 5 red marbles and 5 blue marbles. You randomly draw 1 marble. If it is red, the player receives $8. If it is blue, the player receives $0. What is the expected value (long-run average net gain/loss) for the player for one play, including the $3 cost?
19. A jar contains 50 marbles. Event $A$ is "a randomly selected marble is red," and event $B$ is "a randomly selected marble is large." There are 12 marbles that are both red and large, and there are 20 large marbles total. Based on the information given, what is $P(A\mid B)$? Give your answer as a simplified fraction.
20. A fair six-sided die is rolled once (outcomes $1$ through $6$ are equally likely). Define the random variable $X$ by the mapping: $1\mapsto -2$, $2\mapsto -2$, $3\mapsto 0$, $4\mapsto 0$, $5\mapsto 3$, $6\mapsto 9$. What is the expected value of $X$ based on this theoretical model?

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