Q: A sample of 100 students is taken to estimate the mean number of hours spent on homework. The sample mean is 5 hours, and the sample standard deviation is 2 hours. What is the 95% confidence interval for the mean?

$$5 \pm 1.96 \times \frac{2}{\sqrt{100}}$$

Question 1

Suppose the probability that a randomly selected Math 11 student arrives late for an exam is 0.2. If there are 25 students, and lateness is independent, what is the probability that exactly 5 students are late? Let $$X be $$the number of late students.

Accepted Answer

$$P(X = 5) = \binom{25}{5} (0.2)^5$ (0.8$$)^{20}$$

Question 2

Which of the following is a random variable?

Accepted Answer

The number of heads in 10 tosses of a fair coin

Question 3

Let $$X$$ and $$Y be $$independent random variables with $$E[X] = 2$$, $$E[Y] = 3$$, $$Var(X) = 4$$, and $$Var(Y) = 9. $$What is $$Var(X + Y)$$?

Accepted Answer

$$Var(X + Y) = 13$$

Question 4

Which of the following statements about the expectation operator $$E[\cdot] is $$always true?

Accepted Answer

$$E[aX + b] = aE[X] + b$$ for constants $$a$$ and $$b$$

Question 5

A researcher wants to estimate the average number of hours per week UCSD students spend outside of class on Math 11. Which of the following is the correct population for this study?

Accepted Answer

All UCSD students

Question 6

Suppose $$X \sim 	ext{Uniform}(0, 5). $$What is $$P(X \u003e 3)$$?

Accepted Answer

$$P(X \u003e 3) = \frac{2}{5}$$

Question 7

A 90% confidence interval for the mean number of hours spent per week is $$(3.5166, 3.7516). $$Which of the following is the correct interpretation?

Accepted Answer

We are 90% confident that the true mean number of hours lies between 3.5166 and 3.7516.

Question 8

If $$X \sim 	ext{Poisson}(\lambda)$$, what is $$E[X]$$?

Accepted Answer

$$E[X] = \lambda$$

Question 9

Suppose the time (in days) to grade a Math 11 assignment is uniformly distributed between 1 and 5 days. What is the probability that it takes more than 3 days to grade an assignment?

Accepted Answer

$$P(X \u003e 3) = \frac{2}{4} = 0.5$$

Question 10

Let $$X \sim 	ext{Exp}(\lambda = 0.5). $$What is the probability that $$X \u003e 2$$?

Accepted Answer

$$P(X \u003e 2) = e$$^{-1}$$ \approx 0.3679$$

Question 11

If $$X$$ and $$Y$$ are independent, which of the following is true?

Accepted Answer

$$E[XY] = E[X]E[Y]$$

Question 12

A sample of 100 students is taken to estimate the mean number of hours spent on homework. The sample mean is 5 hours, and the sample standard deviation is 2 hours. What is the 95% confidence interval for the mean?

Accepted Answer

$$5 \pm 1.96 	imes \frac{2}{\sqrt{100}}$$

Question 13

Which of the following is NOT an assumption for constructing a confidence interval for a mean using the $$t-$$distribution?

Accepted Answer

The population standard deviation is known

Question 14

Suppose $$X \sim 	ext{Uniform}(a, b). $$What is the expected value $$E[X]$$?

Accepted Answer

$$E[X] = \frac{a + b}{2}$$

Question 15

A student is told that the number of times they will be called on in class in a week follows a Poisson distribution with mean 2. What is the probability they are called on exactly once?

Accepted Answer

$$P(X = 1) = 2e$$^{-2}$$

Question 16

If $$X \sim 	ext{Exp}(\lambda)$$, what is the memoryless property?

Accepted Answer

$$P(X \u003e s + t \mid X \u003e s) = P(X \u003e t)$$ for all $$s, t \geq 0$$

Question 17

Which of the following is the correct formula for the variance of a sum of two independent random variables $$X$$ and $$Y$$?

Accepted Answer

$$Var(X + Y) = Var(X) + Var(Y)$$

Question 18

Suppose you have already waited 3 days for your assignment to be graded, and the grading time $$X is $$uniformly distributed between 1 and 5 days. What is the probability you have to wait at least 1 more day?

Accepted Answer

$$P(X \u003e 4 \mid X \u003e 3) = \frac{1}{2}$$

Question 19

Which of the following is the correct formula for the probability mass function of a Poisson random variable with parameter $$\lambda$$?

Accepted Answer

$$P(X = k) = \frac{\lambda^k e$$^{-\lambda}$$}{k!}$$

Midterm 2 Study Guide: Probability, Distributions, and Confidence Intervals

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