Q: A box contains 5 red balls and 7 blue balls. If two balls are drawn one after another without replacement, what is the probability that both are red?

$$P = \frac{5}{12} \times \frac{4}{11} = \frac{20}{132} = \frac{5}{33}$$

Q: A fair die is rolled twice. What is the probability that at least one 6 appears?

$$P(\text{at least one 6}) = 1 - P(\text{no 6}) = 1 - (\frac{5}{6})^2$ = 1 - \frac{25}{36} = \frac{11}{36}$$

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

$$E(X) = 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 0 + 0.5 + 0.6 = 1.1$$

Q: In a medical study, 30% of patients experience side effect A, and 20% experience side effect B. If the events are independent, what is the probability that a randomly selected patient experiences both side effects?

$$P(A \cap B) = 0.3 \times 0.2 = 0.06$$

Question 1

A hospital tracks the recovery times (in days) for two different treatments for the same illness. Treatment A has a mean recovery time of 8 days with a standard deviation of 2 days. Treatment B has a mean of 10 days with a standard deviation of 1.5 days. Which treatment shows greater relative variability in recovery times?

Accepted Answer

Treatment A, because its coefficient of variation is higher.

Question 2

A student scores 82 on a statistics exam where the class mean is 75 and the standard deviation is 5. What is the student's z-score?

Accepted Answer

$$z = \frac{82 - 75}{5} = 1.4$$

Question 3

A data set has a five-number summary: 12, 18, 22, 27, 35. What is the interquartile range (IQR)?

Accepted Answer

$$IQR = Q_3$ - Q_1$ = 27 - 18 = 9$$

Question 4

In a probability experiment, the probability of event A is 0.7. What is the probability of the complement of A?

Accepted Answer

$$P(A^c) = 1 - 0.7 = 0.3$$

Question 5

A box contains 5 red balls and 7 blue balls. If two balls are drawn one after another without replacement, what is the probability that both are red?

Accepted Answer

$$P = \frac{5}{12} 	imes \frac{4}{11} = \frac{20}{132} = \frac{5}{33}$$

Question 6

Which of the following is NOT a valid probability distribution?

Accepted Answer

A table where one probability is negative.

Question 7

A fair die is rolled twice. What is the probability that at least one 6 appears?

Accepted Answer

$$P(	ext{at least one 6}) = 1 - P(	ext{no 6}) = 1 - (\frac{5}{6})^2$ = 1 - \frac{25}{36} = \frac{11}{36}$$

Question 8

A student’s height is at the 90th percentile in her class. What does this mean?

Accepted Answer

She is taller than 90% of the students in her class.

Question 9

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.2$$, what is $$P(A \cup B)$$?

Accepted Answer

$$P(A \cup B) = 0.4 + 0.5 - 0.2 = 0.7$$

Question 10

A probability distribution for a random variable X is given by $$P(X=0)=0.2$$, $$P(X=1)=0.5$$, $$P(X=2)=0.3. $$What is the expected value (mean) of X?

Accepted Answer

$$E(X) = 0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 0 + 0.5 + 0.6 = 1.1$$

Question 11

A researcher claims that a value is 'unusual' if it is more than 2 standard deviations from the mean. If a data set has mean 50 and standard deviation 4, which value is considered 'unusual' by the Range Rule of Thumb?

Accepted Answer

A value of 60

Question 12

In a medical study, 30% of patients experience side effect A, and 20% experience side effect B. If the events are independent, what is the probability that a randomly selected patient experiences both side effects?

Accepted Answer

$$P(A \cap B) = 0.3 	imes 0.2 = 0.06$$

Question 13

A boxplot of exam scores for two classes shows that Class A has a larger interquartile range (IQR) than Class B. What does this indicate?

Accepted Answer

Class A has more variability in the middle 50% of scores.

Question 14

If $$P(B|A) = 0.8$$ and $$P(A) = 0.5$$, what is $$P(A \cap B)$$?

Accepted Answer

$$P(A \cap B) = P(B|A) 	imes P(A) = 0.8 	imes 0.5 = 0.4$$

Question 15

A probability distribution has $$P(X=1)=0.3$$, $$P(X=2)=0.4$$, $$P(X=3)=0.2$$, $$P(X=4)=0.1. Is $$this a valid probability distribution?

Accepted Answer

Yes, because all probabilities are between 0 and 1 and sum to 1.

Question 16

A company tests 3 smoke detectors, each with a 0.98 probability of working. What is the probability that at least one detector fails?

Accepted Answer

$$P(	ext{at least one fails}) = 1 - (0.98)^3$ = 1 - 0.941192 = 0.058808$$

Question 17

A sample of 20 students has the following five-number summary for their test scores: 55, 65, 75, 85, 100. Which score is the median?

Accepted Answer

75

Question 18

In a survey, 60% of respondents are female and 40% are male. If a respondent is selected at random, what is the probability that the respondent is not female?

Accepted Answer

$$P(	ext{not female}) = 1 - 0.6 = 0.4$$

Statistics Exam 2 Study Guide: Variability, Relative Standing, Probability, and Probability Distributions

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