Practice Test 22 - Statistics and Probability

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

A coin is claimed to be fair, so $P(\text{heads})=0.5$ . You flip it 20 times and observe 15 heads. You run a simulation of 60 sets of 20 flips assuming a fair coin. In the 60 simulated sets, 11 sets had 15 or more heads ("as extreme or more extreme" means $\ge 15$ heads). Which conclusion is most reasonable based on the simulations?
Using the outcomes listed in $S=\{\text{small},\text{medium},\text{large}\}$ for randomly selecting one shirt size, event $A$ is "not medium." Which set represents $A$ ?
A game pays $30 with probability .15 and $5 with probability .4, else $0. What entry fee makes expected profit zero?
A student is selected at random from a large group. Let event $A$ be "the student prefers tea" and event $B$ be "the student prefers working in the morning." Given that $P(A)=0.45$ , $P(B)=\tfrac{1}{3}$ , and $P(A\cap B)=0.12$ , are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$ .
A scatter plot shows the relationship between the number of practice problems completed (problems) and the score on a unit test (points). The plot shows an upward trend, and technology reports $r=0.74$ for problems completed vs. test score. Which statement best interprets the value of $r$ ?
A number is chosen at random from the set $\{1,2,3,4,5,6,7,8,9,10,11,12\}$ . Event $B$ : the number is a multiple of 3. Event $A$ : the number is even. Using the set as the model, what is $P(A\mid B)$ ? Write your answer as a simplified fraction.
A school district is deciding whether to add a second step to a screening process for identifying students who may need extra reading support. About $12\%$ of students truly need extra support. Current screening (Step 1): correctly flags $85\%$ of students who need support, but also flags $18\%$ of students who do not. Proposed change: Keep Step 1, but require a second short assessment (Step 2) before assigning support. Step 2 correctly confirms $80\%$ of truly-needed students who were flagged by Step 1, and it incorrectly confirms only $10\%$ of truly-not-needed students who were flagged by Step 1. The district's goal is to reduce the number of students incorrectly assigned support while still ensuring that a majority of students who truly need support are assigned it. Which strategy is most reasonable given the probabilities?
School context: Event A is "a student submitted the permission slip," and event B is "a student is going on the field trip." A staff member asks, "Given that a student is going on the field trip, what does 'the chance of A given B' describe?" Which statement best describes what "the chance of A given B" means in this situation?
A random sample of 180 students was surveyed about whether they bring a packed lunch and whether they buy a drink at school. Using the two-way table, what is $P(\text{Packed lunch} \mid \text{Buys a drink})$ ? Give your answer as a decimal. Two-way table (counts): - Rows: Packed lunch (Yes, No) - Columns: Buys a drink (Yes, No) Packed Yes: Drink Yes 36, Drink No 54 (Row total 90) Packed No: Drink Yes 54, Drink No 36 (Row total 90) Column totals: Drink Yes 90, Drink No 90, Grand total 180
A random sample of 110 students recorded whether they attend after-school tutoring and whether they passed the most recent math quiz. Using the two-way table, estimate the probability that a randomly selected student passed given that the student attends tutoring. Give your answer as a percent. Two-way table (counts): - Tutoring & Passed: 32 - Tutoring & Did not pass: 18 - No tutoring & Passed: 45 - No tutoring & Did not pass: 15 Row totals: Tutoring 50, No tutoring 60 Column totals: Passed 77, Did not pass 33 Grand total: 110
A school club sells a lottery-style ticket for $2. One ticket is drawn at random from a box containing 20 tickets: 1 ticket pays $10, 3 tickets pay $4, and the other 16 tickets pay $0. From the player's perspective, what is the expected value (long-run average net gain/loss) of buying 1 ticket?
A factory monitors defects. Event $A$ is "a randomly selected item fails the weight check," and event $B$ is "a randomly selected item fails the color check." Given that $P(A)=0.10$ , $P(B)=0.40$ , and $P(A\cap B)=0.04$ , are $A$ and $B$ independent? Justify using probabilities (compare $P(A\cap B)$ to $P(A)P(B)$ ).
A runner tracked the time (in minutes) to complete a short route over 14 days: 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 25 How does the outlier (25 minutes) affect the mean and the median?
A city is planning snow response for a weekend. Forecast data for the region: - $20\%$ chance of heavy snow - $50\%$ chance of light snow - $30\%$ chance of no snow If the city pre-treats roads Friday night, it reduces the chance of major traffic disruption to: - $15\%$ if heavy snow occurs - $5\%$ if light snow occurs - $1\%$ if no snow occurs If the city does not pre-treat, the chance of major traffic disruption is: - $40\%$ if heavy snow occurs - $12\%$ if light snow occurs - $1\%$ if no snow occurs The city's goal is to minimize the overall probability of major traffic disruption. Which strategy is most reasonable given the probabilities?
A bag contains 3 red marbles and 1 blue marble. One marble is drawn at random. Define the random variable $X$ as the number of red marbles drawn. Which table correctly represents the probability distribution of $X$ ?
A community center wants to estimate the population parameter $p$ , the proportion of all visitors who come for fitness classes. A random sample of $n=150$ visitors was surveyed, and the sample statistic was $\hat p=0.40$ (40%). A simulation repeatedly took 1,000 random samples of size 150 and recorded $\hat p$ each time. The simulation summary said: "About 95% of the simulated sample proportions were within 0.05 of 0.40." Which interval is a reasonable estimate range for the population parameter $p$ ? (Give answers as percents.)
A fair six-sided die is rolled once. Let $X$ be the number of points scored, where you score 1 point if the die shows 1, 2, or 3, and you score 2 points if the die shows 4, 5, or 6. Which table correctly represents the probability distribution of $X$ ?
A club has 6 members who all want to be the discussion leader: P, Q, R, S, T, and U. The club wants a fair method, meaning each person should have probability $\tfrac16$ of being chosen. Which procedure is fair?
A biologist models fish length $y$ (cm) from fish age $x$ (months) using $\hat{y}=6+0.9x$ . A residual plot shows residuals randomly scattered around 0, but with a clear curved trend: residuals are positive at low $x$ , negative in the middle, and positive at high $x$ . What does the residual pattern suggest about the model choice?
A university wants to estimate the mean number of hours per week that all full-time undergraduates spend studying. It randomly selects 80 full-time undergraduates and asks each one to report their study hours for a typical week. Which statement correctly describes what can and cannot be inferred?

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