Practice Test 3 - 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 card is drawn from a standard 52-card deck. Let event $A$ = "the card is a heart" and event $B$ = "the card is a face card (J, Q, or K)." Given that $P(A)=\tfrac{1}{4}$, $P(B)=\tfrac{3}{13}$, and $P(A\cap B)=\tfrac{3}{52}$, are $A$ and $B$ independent? Justify using probabilities by comparing $P(A\cap B)$ to $P(A)P(B)$.
2. A random sample of 180 students recorded whether they take the bus to school and whether they arrive before the first bell. Using the two-way table, what is $P(\text{arrives before bell}\mid\text{takes the bus})$? Give your answer as a decimal. Two-way table (counts): - Bus & Before bell: 64 - Bus & Not before bell: 16 - Not bus & Before bell: 70 - Not bus & Not before bell: 30 Row totals: Bus 80, Not bus 100 Column totals: Before bell 134, Not before bell 46 Grand total: 180
3. A game has two sequential events. First, you roll a fair six-sided die; event $A$ is "the roll is a 6," so $P(A)=\frac{1}{6}$. Then, given that you rolled a 6, you flip a coin that lands heads with probability $P(B\mid A)=\frac{1}{2}$. What is the probability that both events occur, $P(A\cap B)$?
4. A fair six-sided die is rolled once. Define the random variable $X$ as the number of even results (2, 4, or 6) obtained. Which table correctly represents the probability distribution of $X$?
5. A café is choosing between two coupon strategies to distribute. Consider the café's perspective: expected cost per coupon equals the expected discount given. Strategy A (Simple Coupon): - 1.00 probability the customer gets $1 off Strategy B (Spin Wheel): - 0.50 probability the customer gets $0 off - 0.40 probability the customer gets $2 off - 0.10 probability the customer gets $8 off Which strategy has the greater expected value of discount (higher average cost to the café) over many coupons?
6. Based on the sample space shown, $S=\{\text{cat},\text{dog},\text{fish},\text{bird}\}$ for randomly selecting one pet picture, let event $A$ be "the selection is cat or dog." Which set represents $A^c$ (the complement of $A$, meaning not $A$)?
7. Classes A and B have the same median and IQR. A has an extremely high score; B has no extreme scores. Which is true?
8. A teacher recorded the number of pets owned by each of 50 students. The observed counts were: $0$ pets (12 students), $1$ pet (20 students), $2$ pets (13 students), $3$ pets (5 students). Let $X$ be the number of pets for a randomly selected student from this sample. Using the empirical distribution (relative frequencies) as estimated probabilities, what is the expected value of $X$ based on the empirical distribution?
9. A medical screening is done in two stages. First, a person is randomly selected from a group; the probability the person is in the high-risk category is $P(A)=0.15$. Then, given the person is high-risk, the probability the test result is positive is $P(B\mid A)=0.80$. What is the probability that both events occur, $P(A\cap B)$?
10. A delivery company tracks how many packages arrive late each day for 100 days. The observed counts are: 0 late (40 days), 1 late (35 days), 2 late (15 days), 3 late (7 days), 4 late (3 days). Let $X$ be the number of late packages on a randomly selected day. Using the relative frequencies from these 100 days as estimated probabilities, what is the expected value of $X$ based on the empirical distribution?
11. A local news headline says: "Students who spend more time on homework get higher grades—homework time boosts grades!" The article is based on a survey of 500 students comparing hours of homework per week and GPA; students were not assigned homework amounts. Which statement is the most reasonable conclusion?
12. A card is drawn at random from the following 12 equally likely cards: $\{1R, 2R, 3R, 4R, 5R, 6R, 1B, 2B, 3B, 4B, 5B, 6B\}$ (For example, $3R$ means a red 3 and $3B$ means a blue 3.) Event $B$: the card is red. Event $A$: the number on the card is greater than 4. Based on the outcomes listed, which value best represents $P(A\mid B)$? Give your answer as a simplified fraction.
13. A customer at a café randomly receives one of three coupon types: 10% off (probability 0.50), 20% off (probability 0.30), or 50% off (probability 0.20). Define $X$ as the percent discount received. Which table correctly represents the probability distribution of $X$?
14. A random sample of 150 students was surveyed about whether they use a planner and whether they turn in homework on time. Using the two-way table, what is $P(\text{On time} \mid \text{Uses a planner})$? Give your answer as a decimal. Two-way table (counts): - Rows: Uses a planner (Yes, No) - Columns: Homework on time (Yes, No) Planner Yes: On time Yes 60, On time No 15 (Row total 75) Planner No: On time Yes 45, On time No 30 (Row total 75) Column totals: On time Yes 105, On time No 45, Grand total 150
15. A school district wants to estimate the proportion of all district middle-school students who bring a packed lunch at least 3 days per week. The district uses a computer to randomly select 200 middle-school students from the district roster and surveys them; 86 report bringing a packed lunch at least 3 days per week. What is a reasonable inference based on this random sample?
16. A snack company wants to estimate the mean number of crackers in its 1-ounce packages produced today. The company randomly selects 40 packages from today's production line and counts the crackers in each package. What population parameter is being estimated?
17. A wildlife group wants to estimate the population parameter $p$, the proportion of trees in a park that are oak trees. A random sample of $n=100$ trees was selected, and the sample statistic was $\hat p=0.30$ (30%). A simulation repeatedly took 1,000 random samples of size 100 from a detailed map of all trees in the park and recorded $\hat p$ each time. In the simulation, about 95% of the simulated $\hat p$ values were between 0.24 and 0.36. Which interval is a reasonable estimate range for the population parameter $p$?
18. A store used the model $\hat{y}=500+20x$ to predict weekly sales (y, in dollars) from the number of advertisements run (x). The residuals for several weeks are: x: 1, 2, 3, 4, 5, 6 residual: -1, 2, -2, 1, -1, 1 Which statement best describes how well the model fits the data?
19. A bag contains 4 cards labeled $1,2,3,4$. One card is drawn at random; each outcome is equally likely. Define the random variable $X$ by the mapping: $\{1,2\}\mapsto 0$ and $\{3,4\}\mapsto 5$. What is the expected value of $X$ based on this theoretical model?
20. A principal wants to learn whether students who participate in after-school tutoring tend to have higher math grades. The principal collects last semester's math grades for students who chose to attend tutoring and for students who did not, and then compares the two groups. No one is assigned to tutoring by the school. Which statement best describes this study and what (if anything) it allows?

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