Q: A binomial random variable $$X$$ has $$n = 20$$ and $$p = 0.3. $$What is the standard deviation of $$X$$?

$$\sqrt{20 \times 0.3 \times 0.7}$$

Question 1

Which of the following correctly represents the formula for the number of permutations of $$n$$ objects taken $$r at a $$time?

Accepted Answer

$$P(n, r) = \frac{n!}{(n - r)!}$$

Question 2

A researcher wants to select a committee of 4 people from a group of 10. In how many ways can this be done if the order does not matter?

Accepted Answer

$$\frac{10!}{4!\,6!}$$

Question 3

Which formula gives the mean $$\mu of a $$discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$p(x_i)$$?

Accepted Answer

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

Question 4

A fair coin is tossed 5 times. What is the expected number of heads?

Accepted Answer

$$2.5$$

Question 5

Which of the following is the correct formula for the standard deviation $$\sigma of a $$discrete random variable $$X$$?

Accepted Answer

$$\sigma = \sqrt{\sum (x_i - \mu)^2$ p(x_i)}$$

Question 6

If $$X is a $$binomial random variable with parameters $$n$$ and $$p$$, what is the formula for its variance?

Accepted Answer

$$np(1-p)$$

Question 7

A factory produces light bulbs with a 2% defect rate. If 100 bulbs are tested, what is the expected number of defective bulbs?

Accepted Answer

$$2$$

Question 8

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

Accepted Answer

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

Question 9

Suppose $$X is a $$normal random variable with mean $$\mu$$ and standard deviation $$\sigma. $$What is the formula for converting $$X to a $$standard normal variable $$Z$$?

Accepted Answer

$$Z = \frac{X - \mu}{\sigma}$$

Question 10

A standardized test has scores that are normally distributed with mean 500 and standard deviation 100. What is the $$z-$$score for a student who scored 650?

Accepted Answer

$$1.5$$

Question 11

Which of the following best describes the sampling distribution of the sample mean $$\bar{x}$$ for samples of size $$n$$ from a population with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

$$\mu_{\bar{x}} = \mu$$, $$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$

Question 12

A population has mean $$\mu = 80$$ and standard deviation $$\sigma = 12. $$What is the standard deviation of the sample mean for samples of size $$n = 16$$?

Accepted Answer

$$3$$

Question 13

A binomial random variable $$X$$ has $$n = 20$$ and $$p = 0.3. $$What is the standard deviation of $$X$$?

Accepted Answer

$$\sqrt{20 	imes 0.3 	imes 0.7}$$

Question 14

A quality control manager samples 25 items from a production line where the defect rate is 4%. What is the probability that exactly 1 item is defective? (Use the binomial formula.)

Accepted Answer

$$\binom{25}{1} (0.04)^1$ (0.96$$)^{24}$$

Question 15

If a random variable $$X$$ has mean $$\mu$$ and variance $$\$$sigma^2$$, what is the variance of the sample mean $$\bar{x}$$ for samples of size n$?

Accepted Answer

$$\frac{\$$sigma^2$$}{n}$$

Question 16

A student randomly selects 3 books from a shelf of 8 different books. In how many ways can this be done if the order matters?

Accepted Answer

$$\frac{8!}{(8-3)!}$$

Question 17

Which of the following statements about the normal distribution is TRUE?

Accepted Answer

Approximately 68% of the data falls within one standard deviation of the mean.

Question 18

A population has $$\mu = 50$$ and $$\sigma = 10. $$What is the probability that a randomly selected value is more than 2 standard deviations above the mean (assuming normality)?

Accepted Answer

Approximately 2.5%

Question 19

If $$X is a $$binomial random variable with $$n = 10$$ and $$p = 0.5$$, what is $$P(X = 5)$$?

Accepted Answer

$$\binom{10}{5} (0.5)^5$ (0.5)^5$

Permutations, Combinations, and Distributions in Statistics: Study Notes

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