Q: Suppose a random variable $$X$$ has the following probability distribution: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3. $$What is the mean (expected value) of $$X$$?

$$E(X) = 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1$$

Q: A binomial distribution has $$n = 8$$ trials and probability of success $$p = 0.25. $$What is the mean number of successes?

$$\mu = np = 8 \times 0.25 = 2$$

Q: A data set has $$Q_1 = 15$, Q_3$ = 25$$, and $$IQR = 10. $$Which value would be considered an outlier using the $$1.5 \times IQR$$ rule?

A value less than $$15 - 1.5 \times 10 = 0 or $$greater than $$25 + 1.5 \times 10 = 40.$$

Q: A binomial distribution has $$n = 5$$ and $$p = 0.6. $$What is the standard deviation?

$$\sigma = \sqrt{npq} = \sqrt{5 \times 0.6 \times 0.4} = \sqrt{1.2} \approx 1.095$$

Question 1

A sample of exam scores is: 72, 85, 90, 68, 95. What is the range of this data set?

Accepted Answer

$$95 - 68 = 27$$

Question 2

Given a data set with mean $$\mu = 50$$ and standard deviation $$\sigma = 5$$, what is the z-score for a value of $$60$$?

Accepted Answer

$$z = \frac{60 - 50}{5} = 2$$

Question 3

A data set has a mean of $$100$$ and a standard deviation of $$15. $$According to the empirical rule, approximately what percentage of data falls between $$85$$ and $$115$$?

Accepted Answer

About 68%

Question 4

Which of the following is NOT a property of probabilities?

Accepted Answer

A probability can be negative.

Question 5

A boxplot is constructed from the following five-number summary: Min = 10, Q1 = 20, Median = 30, Q3 = 40, Max = 50. What is the interquartile range (IQR)?

Accepted Answer

$$IQR = Q_3$ - Q_1$ = 40 - 20 = 20$$

Question 6

A fair die is rolled. What is the probability of rolling a number greater than 4?

Accepted Answer

$$P = \frac{2}{6} = \frac{1}{3}$$

Question 7

Which of the following is a requirement for a probability distribution of a discrete random variable $$X$$?

Accepted Answer

$$\sum P(x) = 1$$

Question 8

Suppose a random variable $$X$$ has the following probability distribution: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3. $$What is the mean (expected value) of $$X$$?

Accepted Answer

$$E(X) = 0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 1.1$$

Question 9

Which of the following scenarios describes a binomial experiment?

Accepted Answer

Flipping a coin 10 times and counting the number of heads.

Question 10

A binomial distribution has $$n = 8$$ trials and probability of success $$p = 0.25. $$What is the mean number of successes?

Accepted Answer

$$\mu = np = 8 	imes 0.25 = 2$$

Question 11

A continuous uniform distribution has minimum value $$a = 2$$ and maximum value $$b = 8. $$What is the probability that a randomly selected value is between $$3$$ and $$5$$?

Accepted Answer

$$P = \frac{5 - 3}{8 - 2} = \frac{2}{6} = \frac{1}{3}$$

Question 12

If a value has a z-score of $$-2.5 in a $$standard normal distribution, what does this indicate?

Accepted Answer

The value is 2.5 standard deviations below the mean.

Question 13

A data set has $$Q_1 = 15$, Q_3$ = 25$$, and $$IQR = 10. $$Which value would be considered an outlier using the $$1.5 	imes IQR$$ rule?

Accepted Answer

A value less than $$15 - 1.5 	imes 10 = 0 or $$greater than $$25 + 1.5 	imes 10 = 40.$$

Question 14

If the probability of rain on a given day is $$0.3$$, what is the probability that it does NOT rain?

Accepted Answer

$$1 - 0.3 = 0.7$$

Question 15

A standard normal distribution has mean $$0$$ and standard deviation $$1. $$What is the probability that a randomly selected value is less than $$z = 1$$? (Use the empirical rule.)

Accepted Answer

About 84%

Question 16

A probability distribution is given by $$P(X=1) = 0.4$$, $$P(X=2) = 0.3$$, $$P(X=3) = 0.2$$, $$P(X=4) = 0.1. $$What is the standard deviation of $$X$$?

Accepted Answer

$$\sigma = \sqrt{\sum (x - \mu)^2$ P(x)} = \sqrt{0.76} \approx 0.87$$

Question 17

A survey asks 100 people if they have ever traveled abroad. If 30 people say yes, what is the probability that a randomly selected person from this group has traveled abroad?

Accepted Answer

$$P = \frac{30}{100} = 0.3$$

Question 18

A binomial distribution has $$n = 5$$ and $$p = 0.6. $$What is the standard deviation?

Accepted Answer

$$\sigma = \sqrt{npq} = \sqrt{5 	imes 0.6 	imes 0.4} = \sqrt{1.2} \approx 1.095$$

Question 19

If a value is more than 3 standard deviations above the mean in a normal distribution, what can you conclude?

Accepted Answer

The value is considered significantly high (an outlier).

MAT 177 Test 2 Study Guide: Variation, Probability, and Distributions

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