Using the table above summarizing test‐taking strategies and scores for 1000 students, what is the probability that a randomly chosen student used the “Guess” strategy?

A company manufactures batteries with lifetimes (in years) that follow a discrete random variable Y with the following distribution: P(Y = 3) = 0.25, P(Y = 4) = 0.45, P(Y = 5) = 0.30. If the company offers a warranty to replace any battery that fails before its expected lifetime, what is the warranty period?

In the previous question, the probability that a student uses a guessing strategy is $$0.3$$. However, if only $$0.25$$ of students who score below 60 used guessing, are the events "using guessing" and "scoring below 60" independent?

What is a key characteristic of a binomial distribution?

Which of the following is true regarding the joint probability that a test-taker chooses 'answer (c)' and scores between 80 and 100, if 70 out of 1000 test-takers meet this criterion?

When analyzing data on household income and educational attainment across different regions, a statistician notices that the positive association is stronger in urban areas than in rural areas. What question is most appropriately suggested by this differential pattern?

I. If the probability that a test taker uses the guess strategy is $$0.3$$ and the probability of scoring 0–59 is $$0.2$$, then assuming independence the joint probability would be $$0.3 \times 0.2 = 0.06$$. II. If the conditional probability $$P(\text{guess} | \text{score 0–59})$$ does not equal $$P(\text{guess})$$, then the two events are not independent. III. Two events are independent if and only if the conditional probability of one given the other equals its marginal probability. Which of the above statements is/are true?

Which equation represents the independence of events A and B?

An insurance salesperson estimates the daily numbers of auto and home insurance policies sold using independent probability distributions. A student calculates the overall variance in policies sold by averaging the individual variances instead of adding them. What is the mistake in the student's calculation?

Two students perform simulations to estimate the probability of a complex event. Student A performs 100 trials and gets an estimate of 0.43. Student B performs 10,000 trials and gets an estimate of 0.412. The true probability is unknown. Which statement is most justified?

A lottery sells tickets for $1 with a prize of $7500 for one winner (the net gain for the winner is $7499). Based on the table above, what is the expected net outcome per ticket?

The cumulative distribution function of a continuous random variable X is given by F(x) = (x²/16) for 0≤x≤4. What is P(1<X≤3)?

Which of the following is true regarding the formula for the variance of a discrete random variable?

An email survey is sent out to a large group of individuals, but only 40% of those invited respond. If the nonrespondents differ significantly from respondents in key characteristics, which type of bias is most likely affecting the survey results?