Q: If $$X is a $$binomial random variable with $$n=5$$ and $$p=0.4$$, what is the probability of exactly 3 successes?

$$P(X=3) = 10 \times (0.4)^3$ \times (0.6)^2$

Question 1

Which of the following best describes a random variable?

Accepted Answer

A variable whose value is determined by chance and uncertainty in a random experiment.

Question 2

Suppose a family plans to have four children. What is the probability that exactly two of them are girls, assuming the probability of a girl is 0.5?

Accepted Answer

$$P(X=2) = \frac{6}{16}$$

Question 3

Which of the following is NOT a property of the normal distribution?

Accepted Answer

It is always skewed to the right.

Question 4

A box contains 5 defective and 10 non-defective clocks. If three clocks are selected at random without replacement, what is the probability that none are defective?

Accepted Answer

$$\frac{120}{455}$$

Question 5

Which of the following is the formula for the mean of a discrete random variable $$X$$ with probability mass function $$p(x)$$?

Accepted Answer

$$\mu = \sum x p(x)$$

Question 6

If $$X is a $$binomial random variable with $$n=5$$ and $$p=0.4$$, what is the probability of exactly 3 successes?

Accepted Answer

$$P(X=3) = 10 	imes (0.4)^3$ 	imes (0.6)^2$

Question 7

Which of the following sampling techniques divides the population into groups and then randomly selects some groups to sample all members within them?

Accepted Answer

Cluster sampling

Question 8

Given a probability density function $$f(x) = kx$$ for $$0 \u003c x \u003c 2$$, what is the value of $$k so $$that $$f(x) is a $$valid pdf?

Accepted Answer

$$k = \frac{1}{2}$$

Question 9

A researcher wants to estimate the average milk production of cows in a region. Which sampling method ensures every cow has an equal chance of being selected?

Accepted Answer

Simple random sampling

Question 10

If the probability of a rare disease in a population is 0.001, and 1000 people are tested, which distribution best models the number of cases found?

Accepted Answer

Poisson distribution

Question 11

Which of the following is the formula for the standard deviation of the sample mean when sampling from a finite population?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}}$$

Question 12

A fair coin is tossed three times. What is the expected number of heads?

Accepted Answer

$$1.5$$

Question 13

Which of the following is a property of probability?

Accepted Answer

$$0 \leq P(A) \leq 1$$

Question 14

If $$A$$ and $$B$$ are mutually exclusive events, what is $$P(A \cap B)$$?

Accepted Answer

$$0$$

Question 15

A student answers a 6-question multiple-choice exam, each with 4 possible answers. How many different answer keys are possible?

Accepted Answer

$$4^6$$

Question 16

Which of the following best describes empirical (relative frequency) probability?

Accepted Answer

Probability based on observed data from experiments or past events.

Question 17

If $$X is a $$normal random variable with mean $$\mu = 80$$ and standard deviation $$\sigma = 4.8$$, what is the $$Z-$$score for $$X = 72$$?

Accepted Answer

$$Z = \frac{72-80}{4.8} = -1.67$$

Question 18

Which of the following is an example of a continuous random variable?

Accepted Answer

The amount of milk produced by a cow in a day.

Question 19

Which of the following is the formula for the probability of $$k$$ successes in $$n$$ independent Bernoulli trials with probability $$p of $$success?

Accepted Answer

$$P(X=k) = {n \choose k} p^k (1-p$$)^{n-k}$$

Probability, Random Variables, and Sampling Distributions: Structured Study Notes for Statistics

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