Practice Test 49 - 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 bag contains 2 red marbles (R) and 1 blue marble (B). One marble is drawn at random. The sample space is $\{R_1, R_2, B\}$. Define the random variable $X$ as the number of blue marbles drawn. Which table correctly represents the probability distribution of $X$?
2. A streaming service surveyed 250 users about how many ads they saw in the first 10 minutes of a show. Observed counts: 0 ads (40 users), 1 ad (85 users), 2 ads (90 users), 3 ads (35 users). Let $X$ be the number of ads seen by a randomly selected surveyed user. Using the relative frequencies from the survey as estimated probabilities, what is the expected value of $X$ based on the empirical distribution?
3. A student is selected at random from a large group. Let event $A$ = "the student prefers tea," and event $B$ = "the student prefers coffee." Given that $P(A)=0.30$, $P(B)=0.50$, and $P(A\cap B)=0$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
4. A university reports $P(A)=0.30$, $P(B)=0.50$, and $P(A\cap B)=0.12$, where event $A$ is "a student studies abroad" and event $B$ is "a student is in the honors program." Based on the information given, are $A$ and $B$ independent? Use conditional probability by comparing $P(A\mid B)$ and $P(A)$.
5. Two gardens measured the mass (in grams) of tomatoes harvested on the same day. Data Set A: 140, 145, 148, 150, 152, 153, 155, 156, 158, 160, 162 Data Set B: 120, 130, 140, 145, 150, 152, 154, 155, 156, 158, 210 Which statement best compares the typical value and variability of Data Set A and Data Set B? Use measures appropriate to the distribution shapes.
6. Daily high temperatures (°F) over 14 days were: 62, 64, 65, 65, 66, 66, 67, 68, 68, 69, 70, 71, 72, 74. Which statement correctly describes the distribution shown by a box plot of these temperatures?
7. A card is drawn from a standard 52-card deck without replacement, and then a second card is drawn. Let $A$ be the event "the first card is a heart," and let $B$ be the event "the second card is a heart." Given that the first card was a heart, $P(B\mid A)=\frac{12}{51}$. What is the probability that both events occur, $P(A\cap B)$?
8. A school tracked whether students usually eat breakfast and whether they arrive to first period on time. The results are shown in the two-way table. What is the conditional relative frequency (as a percent) of students who arrive on time given that the student skips breakfast?

   	On Time	Late	Total
   Eats Breakfast	70	30	100
   Skips Breakfast	30	50	80
   Total	100	80	180

9. The diameters of ball bearings produced by a machine are monitored and appear roughly normal (symmetric, bell-shaped). Diameters are modeled as $N(\mu=10.0\text{ mm},\,\sigma=0.2\text{ mm})$. Approximately what percent of bearings have diameter between $9.6$ mm and $10.4$ mm? (Use the empirical rule.)
10. A streaming service models monthly data use with $y = 3.2x + 5,$ where $x$ is the number of hours streamed in a month (hours) and $y$ is the predicted data used (gigabytes). Which interpretation of the y-intercept is correct (include units)?
11. A normal data set has mean 100 and SD 15. Estimate the percentage of values from 85 to 130.
12. A student council collected data on whether students prefer school events on weekdays or weekends and whether they are involved in student government. The results are shown in the two-way table. What is the conditional relative frequency (as a decimal) of students who prefer weekends given that the student is involved in student government?

    	Prefer Weekdays	Prefer Weekends	Total
    Student Government: Yes	18	42	60
    Student Government: No	55	35	90
    Total	73	77	150

13. A cereal company wants to estimate the mean weight of cereal in all boxes produced on a particular day. From the day's production run, the company randomly selects 50 boxes and measures the cereal weight in each box, then computes the sample mean. What population parameter is being estimated?
14. An engineer fits a linear model $\hat{y}=15+0.5x$ to predict battery life $y$ (hours) from screen brightness setting $x$ (percent). The residual plot shows residuals randomly scattered around the 0 line, with roughly equal numbers above and below 0 and no obvious curve. Which statement best describes how well the model fits the data?
15. A fair coin is flipped 3 times. Define the random variable $X$ as the number of heads obtained. Which table correctly represents the probability distribution of $X$?
16. In a classroom wager, you pay $3 to draw one marble from a bag of 10 marbles. If you draw a red marble (4 marbles), you receive $7. If you draw a blue marble (6 marbles), you receive $0. From the player's perspective, what is the expected value (long-run average net gain/loss) of one draw?​
17. Five different books (A, B, C, D, E) are placed in a row in a random order. Order matters because different arrangements count as different outcomes. There are $5!$ equally likely arrangements. What is the probability that books A and B are adjacent (next to each other)?
18. A factory reports $P(A\cap B)=0.08$ and $P(B)=0.16$, where event $A$ is "an item is defective" and event $B$ is "an item came from Machine 2." Based on the information given, what is $P(A\mid B)$? Give your answer as a decimal.
19. The weights of sealed coffee bags from a stable process are approximately normal with mean $\mu=16.0\text{ oz}$ and standard deviation $\sigma=0.2\text{ oz}$. Approximately what percent of bags weigh more than 16.4 oz? (Use the empirical rule.)
20. A spinner has 4 equal sections labeled A, B, C, and D. You spin once. Define the random variable $X$ by: $X=2$ if the result is A or B, $X=5$ if the result is C, and $X=8$ if the result is D. Which table correctly represents the probability distribution of $X$?

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