Q: If two events A and B are independent, what is the probability that both A and B occur?

$$P(A \text{ and } B) = P(A) \times P(B)$$

Question 1

Which of the following is an example of continuous data?

Accepted Answer

Temperature measured in degrees Celsius

Question 2

A variable is defined as:

Accepted Answer

A series of observations that vary across observations

Question 3

When summarizing discrete data, which visualization is most appropriate?

Accepted Answer

Bar chart

Question 4

A pie chart is used to show the percentage of high school students engaging in dangerous behaviors. What is a common mistake when using pie charts?

Accepted Answer

Violating the area principle by making the area of slices not proportional to the percentages

Question 5

Which measure of central tendency is most appropriate for a skewed distribution with outliers?

Accepted Answer

Median

Question 6

If a distribution is right-skewed, how does the mean compare to the median?

Accepted Answer

Mean is higher than the median

Question 7

Suppose two companies have the same average delivery time, but Company A has a much lower variance than Company B. What does this imply?

Accepted Answer

Company A is more consistent in delivery times than Company B

Question 8

The sample variance $$s^2 is $$calculated as:

Accepted Answer

$$s^2$$ = \frac{\sum (y_i - \bar{y}$$)^2$$}{n-1}$$

Question 9

According to the empirical rule, approximately what percentage of observations fall within two standard deviations of the mean in a unimodal, symmetric distribution?

Accepted Answer

95%

Question 10

Which of the following statements about probability is TRUE?

Accepted Answer

The probability of an event is the proportion of times the event would occur in a very long series of repetitions

Question 11

If two events A and B are independent, what is the probability that both A and B occur?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) 	imes P(B)$$

Question 12

In a four-zero roulette game (numbers 1-36 plus four zeros), what is the probability of landing on a red number?

Accepted Answer

$$\frac{18}{40}$$

Question 13

If the probability of buying 5 or more tickets in a transaction is 0.15, what is the probability that two independent transactions are both for 5 or more tickets?

Accepted Answer

$$0.0225$$

Question 14

The expected value of a discrete random variable $$X is $$calculated as:

Accepted Answer

$$E(X) = \sum x_i P(X = x_i)$$

Question 15

If the expected payout per spin on a slot machine is $$0.95 and the standard deviation is 2.05$$, what is the expected payout and standard deviation if the payout amounts are doubled?

Accepted Answer

Question 16

If $$X$$ and $$Y$$ are independent random variables, each with mean $$\mu$$ and variance $$\$$sigma^2$$, what is the mean and variance of Z = X + Y$?

Accepted Answer

Mean: $$2\mu$$, Variance: $$2\$$sigma^2$$

Question 17

A financial portfolio holds 70% in bonds (mean return 0.04, SD 0.046) and 30% in stocks (mean return 0.115, SD 0.199). Assuming independence, what is the expected return and standard deviation of the portfolio?

Accepted Answer

Expected return: $$0.065$$, Standard deviation: $$0.068$$

Question 18

According to the Law of Large Numbers, as the number of observations increases, the sample mean:

Accepted Answer

Converges to the population mean

Question 19

If you subtract a constant $$c$$ from every value of a random variable $$X$$, how does this affect the mean and variance?

Accepted Answer

Mean decreases by $$c$$, variance remains unchanged

Describing Data, Probability, and Random Variables: Study Notes for Business Statistics

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