Practice Test 31 - Statistics and Probability

20 questions. Answer them in any order, then submit to see your score and every explanation.

A plant grows according to the model $y = 3.5x + 12,$ where $x$ is time since planting (weeks) and $y$ is the plant's height (centimeters). Which interpretation of the slope is correct? (Include units.)
A teacher makes a scatter plot of the number of pages read (x) and the number of points earned on a reading log (y). Two representative points on a reasonable line of best fit are marked at $(10,45)$ and $(30,85)$ . Estimate the slope of a reasonable line of best fit (in points per page).
Five volunteers—Kai, Lina, Mo, Nia, and Omar—must be chosen for a single job. A procedure is fair if each person has probability $\tfrac15$ of being selected. Which procedure is fair?
A principal randomly assigned 40 classrooms to two ways of organizing a 10-minute daily reading time for one month: Treatment A (silent independent reading) and Treatment B (teacher read-aloud). Twenty classrooms were randomly assigned to each method. The outcome was the mean increase in a reading fluency score (points) from pretest to posttest. Treatment A had a mean increase of 12.3 points and Treatment B had a mean increase of 10.9 points, so the observed difference in means was $12.3-10.9=1.4$ points (A − B). Under "no treatment effect," the principal shuffled the treatment labels across classrooms 2000 times and recalculated the difference each time. In the simulations, 52 of the 2000 shuffled differences were at least as large as 1.4 (A − B). Which conclusion is most reasonable about the treatment effect?
A school reports: event $A$ is "a randomly selected student takes an art class," and event $B$ is "a randomly selected student takes a technology class." Given that $P(A)=0.35$ , $P(B)=\tfrac{2}{5}$ , and $P(A\cap B)=0.12$ , are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$ .
A runner's coach models the time to complete a run with $y = 12 + 0.8x,$ where $x$ is the distance run (miles) and $y$ is the time (minutes). What does the slope represent in this context? Include units in your interpretation.
A fair coin is flipped three times. Define the random variable $X$ as the number of tails obtained. Which table correctly represents the probability distribution of $X$ ?
A carnival game claims its coin is fair, with $P(\text{heads})=0.5$ . You flip the coin 20 times and get 18 heads. To check whether this seems consistent with the claim, you simulate 100 runs of 20 fair-coin flips. In the simulations, 0 out of 100 runs produced 18 or more heads. Would this result cause you to question the model? Why?
A box contains 3 red balls and 1 blue ball. One ball is drawn at random. Let $X$ be the value of the draw, where $X=2$ if the ball is red and $X=8$ if the ball is blue. The distribution is: $P(X=2)=\tfrac{3}{4}$ , $P(X=8)=\tfrac{1}{4}$ . What is the expected value of $X$ ? (Compute $\sum x\,P(x)$ ; the expected value may not be a possible outcome.)
A spinner is spun once. Event $A$ is "the spinner lands on red," and event $B$ is "the spinner lands on an even-numbered section." Given that $P(A)=0.40$ , $P(B)=0.50$ , and $P(A\cap B)=0.30$ , are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$ .
Daily output from a manufacturing line (in units) is roughly symmetric and bell-shaped over long periods, so a normal model is appropriate. Output has mean $\mu=500$ and standard deviation $\sigma=50$ . Approximately what percent of days have output between 450 and 550 units? (Use the empirical rule.)
A bag contains 9 identical tiles labeled with the following outcomes: $\{A1, A2, A3, B1, B2, B3, C1, C2, C3\}$ (For example, $B2$ means letter $B$ and number 2.) One tile is drawn at random. Event $B$ : the number is 2 or 3. Event $A$ : the letter is $A$ . What fraction of outcomes in $B$ also belong to $A$ ? Give your answer as a simplified fraction.
A researcher recorded the number of hours of sleep (hours) each student reported the night before a test and the student's test score (points). Using technology on the paired data, the correlation coefficient was found to be $r = 0.82$ . Which statement best interprets the value of $r$ for the relationship between hours of sleep and test score?
A fair six-sided die is rolled once. Define the random variable $X$ as the number of points you earn, where you earn 0 points if you roll 1 or 2, 1 point if you roll 3 or 4, and 2 points if you roll 5 or 6. Which table correctly represents the probability distribution of $X$ ?
A password is formed by choosing one letter from $\{A,B,C,D\}$ and then choosing one digit from $\{1,2,3\}$ . All 12 outcomes are equally likely: $\{A1,A2,A3,B1,B2,B3,C1,C2,C3,D1,D2,D3\}$ Event $B$ : the digit is 1. Event $A$ : the letter is $C$ or $D$ . Based on the outcomes listed, which value best represents $P(A\mid B)$ ? Give your answer as a simplified fraction.
To learn whether a new note-taking template helps students remember vocabulary, a teacher has all students use the template for 2 weeks and then compares their vocabulary quiz scores to the class's quiz scores from the previous unit (when the template was not used). No random assignment is used. Which statement is most accurate?
A city reports that neighborhoods with higher average household income tend to have higher average home prices (a positive association). The report used existing census and real-estate data; no neighborhoods were assigned an income level. Which statement is the most reasonable conclusion?
A school randomly selects one student. Event $A$ is "the student takes art," and event $B$ is "the student plays a sport." Given that $P(A)=0.35$ , $P(B)=0.20$ , and $P(A\cap B)=0.07$ , are $A$ and $B$ independent? Justify using the product rule by comparing $P(A\cap B)$ to $P(A)P(B)$ .
A school counselor wants to estimate the population parameter $p$ , the proportion of all seniors who have completed a college application. Two different random samples are considered: - Sample 1: $n=100$ , sample statistic $\hat{p}=0.62$ - Sample 2: $n=400$ , sample statistic $\hat{p}=0.62$ Students run simulations for each sample size. For $n=100$ , about 95% of simulated $\hat{p}$ values are within 0.10 of 0.62. For $n=400$ , about 95% of simulated $\hat{p}$ values are within 0.05 of 0.62. Which interval is a reasonable estimate range for the population parameter $p$ using Sample 2?
A fair coin is flipped 3 times; all $2^3=8$ outcomes are equally likely. Define the random variable $X$ as the number of heads. The outcomes map to $X$ as follows: $\text{TTT}\to 0$ ; $\text{HTT, THT, TTH}\to 1$ ; $\text{HHT, HTH, THH}\to 2$ ; $\text{HHH}\to 3$ . Which table correctly represents the theoretical probability distribution of $X$ ?

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