Q: If events A and B are independent, which formula gives the probability of both A and B occurring?

$$ P(A \cap B) = P(A) \times P(B) $$

Question 1

Which of the following best describes the difference between a parameter and a statistic?

Accepted Answer

A parameter is a numerical value summarizing a characteristic for a population, while a statistic summarizes a characteristic for a sample.

Question 2

A pharmaceutical company wants to estimate the average recovery time for a new drug. They collect data from 200 patients who used the drug. In this context, what is the sample?

Accepted Answer

The 200 patients who used the drug.

Question 3

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

Accepted Answer

The body temperature of a patient measured in degrees Celsius.

Question 4

Suppose you want to avoid bias when selecting a sample from a population. Which method should you use?

Accepted Answer

Random sampling, where every member of the population has an equal chance of being selected.

Question 5

Which measure of center is most resistant to outliers?

Accepted Answer

Median

Question 6

Given the following data set: 2, 4, 4, 6, 8, what is the mean?

Accepted Answer

The mean is $$ \frac{2+4+4+6+8}{5} = \frac{24}{5} = 4.8 $$

Question 7

A boxplot is most useful for which of the following purposes?

Accepted Answer

Summarizing data using the five-number summary (minimum, first quartile, median, third quartile, maximum).

Question 8

If the interquartile range (IQR) of a data set is 12, and the first quartile ($$ Q_1$ ) is 18$$, what is the third quartile ($$ Q_3$ $$)?

Accepted Answer

$$ Q_3$ = Q_1$ + IQR = 18 + 12 = 30 $$

Question 9

Which of the following statements about histograms is correct?

Accepted Answer

Histograms show the distribution of numerical data by grouping data into bins.

Question 10

A probability experiment consists of flipping a fair coin three times. What is the probability of getting exactly two heads?

Accepted Answer

$$ P = \frac{3}{8} $$

Question 11

Which of the following is an example of mutually exclusive events?

Accepted Answer

Drawing a red card and drawing a black card from a standard deck in a single draw.

Question 12

If events A and B are independent, which formula gives the probability of both A and B occurring?

Accepted Answer

$$ P(A \cap B) = P(A) 	imes P(B) $$

Question 13

A medical test for a disease has a 95% chance of correctly identifying a diseased patient (true positive) and a 5% chance of a false positive. If 1% of the population has the disease, what statistical concept should be used to update the probability that a patient has the disease after testing positive?

Accepted Answer

Bayes' Rule

Question 14

Which of the following is NOT a requirement for a binomial experiment?

Accepted Answer

The probability of success must change from trial to trial.

Question 15

If a random variable $$ X $$ can take values 1, 2, and 3 with probabilities 0.2, 0.5, and 0.3 respectively, what is the expected value $$ E[X] $$?

Accepted Answer

$$ E[X] = 1 	imes 0.2 + 2 	imes 0.5 + 3 	imes 0.3 = 0.2 + 1.0 + 0.9 = 2.1 $$

Question 16

A researcher wants to compare the test scores of students from two different schools. The mean score at School A is 75 with a standard deviation of 5, and at School B is 80 with a standard deviation of 10. A student scores 85 at School A and 90 at School B. Which student performed better relative to their school, based on z-scores?

Accepted Answer

School A: $$ z = \frac{85-75}{5} = 2 $$; School B: $$ z = \frac{90-80}{10} = 1 . $$The student at School A performed better relative to their school.

Question 17

Which of the following formulas correctly represents conditional probability?

Accepted Answer

$$ P(A|B) = \frac{P(A \cap B)}{P(B)} $$

Question 18

A bar chart is most appropriate for which type of data?

Accepted Answer

Categorical data

Question 19

If you roll two fair six-sided dice, what is the probability that both dice show a 6?

Accepted Answer

$$ P = \frac{1}{6} 	imes \frac{1}{6} = \frac{1}{36} $$

Fundamental Concepts in Statistics: Populations, Data, Probability, and Binomial Experiments

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