Q: If a data set has a minimum value of 10, a maximum value of 50, and is to be divided into 8 classes, what is the class width?

$$\text{Class width} = \frac{50 - 10}{8} = 5$$

Q: If the probability of rain on a given day is 0.3 and the probability of a traffic jam is 0.4, and the two events are independent, what is the probability that it rains and there is a traffic jam?

$$P(A \cap B) = P(A) \cdot P(B) = 0.3 \times 0.4 = 0.12$$

Q: Given a data set with a minimum value of 5 and a maximum value of 25, what is the midrange?

$$\text{Midrange} = \frac{5 + 25}{2} = 15$$

Question 1

A researcher collects the following data set: 4, 8, 6, 10, 12. What is the sample mean $$\bar{x} of $$this data set?

Accepted Answer

$$\bar{x} = \frac{4 + 8 + 6 + 10 + 12}{5} = 8$$

Question 2

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

Accepted Answer

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

Question 3

A hospital records the number of patients admitted each day for a week: 12, 15, 13, 14, 16, 15, 14. What is the sample standard deviation $$s of $$the data?

Accepted Answer

$$s = \sqrt{\frac{\sum (x_i - \bar{x})^2$}{n-1}} = 1.49$$

Question 4

If a data set has a minimum value of 10, a maximum value of 50, and is to be divided into 8 classes, what is the class width?

Accepted Answer

$$	ext{Class width} = \frac{50 - 10}{8} = 5$$

Question 5

A password consists of 4 distinct letters chosen from the English alphabet (26 letters). How many different passwords are possible?

Accepted Answer

$$nPr = \frac{26!}{(26-4)!} = 358800$$

Question 6

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.2$$, what is the conditional probability $$P(A|B)$$?

Accepted Answer

$$P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.2}{0.5} = 0.4$$

Question 7

A data set has $$Q_1 = 20$, Q_3$ = 40. $$What is the interquartile range (IQR)?

Accepted Answer

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

Question 8

According to the Empirical Rule, approximately what percentage of data falls within two standard deviations of the mean in a normal distribution?

Accepted Answer

About 95%

Question 9

A company wants to form a committee of 3 people from a group of 10 employees. How many different committees can be formed?

Accepted Answer

$$nCr = \frac{10!}{3! \cdot (10-3)!} = 120$$

Question 10

If the probability of rain on a given day is 0.3 and the probability of a traffic jam is 0.4, and the two events are independent, what is the probability that it rains and there is a traffic jam?

Accepted Answer

$$P(A \cap B) = P(A) \cdot P(B) = 0.3 	imes 0.4 = 0.12$$

Question 11

Given a data set with a minimum value of 5 and a maximum value of 25, what is the midrange?

Accepted Answer

$$	ext{Midrange} = \frac{5 + 25}{2} = 15$$

Question 12

A student scores 85 on a test where the mean is 80 and the standard deviation is 5. What is the student's z-score?

Accepted Answer

$$z = \frac{85 - 80}{5} = 1$$

Question 13

If $$P(A) = 0.6$$ and $$P(B) = 0.5$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A \cup B)$$?

Accepted Answer

$$P(A \cup B) = P(A) + P(B) = 1.1$$

Question 14

A data set has $$Q_1 = 15$, Q_3$ = 35$$, and $$IQR = 20. $$What is the lower fence (LF) for detecting outliers?

Accepted Answer

$$LF = Q_1$ - 1.5 	imes IQR = 15 - 1.5 	imes 20 = -15$$

Question 15

A pharmaceutical company tests a new drug on 1000 patients. If the results are normally distributed, how many patients would you expect to have results within one standard deviation of the mean, according to the Empirical Rule?

Accepted Answer

Approximately 680 patients

Question 16

If $$P(A) = 0.7$$, $$P(B) = 0.2$$, and $$P(A \cap B) = 0.1$$, what is $$P(A \cup B)$$?

Accepted Answer

$$P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.7 + 0.2 - 0.1 = 0.8$$

Question 17

A teacher wants to select 2 students from a class of 5 to present a project. In how many ways can the teacher select the students if the order does not matter?

Accepted Answer

$$nCr = \frac{5!}{2! \cdot (5-2)!} = 10$$

Question 18

A data set has a mean of 100 and a standard deviation of 15. What range of values would include approximately 99.7% of the data according to the Empirical Rule?

Accepted Answer

$$100 \pm 3 	imes 15 = 55 	ext{ to } 145$$

Question 19

If the probability that a randomly selected person is left-handed is 0.1 and the probability that a person has blue eyes is 0.2, what is the probability that a person is either left-handed or has blue eyes, assuming the events are independent?

Accepted Answer

$$P(A \cup B) = P(A) + P(B) - P(A)P(B) = 0.1 + 0.2 - 0.02 = 0.28$$

Question 20

A data set has $$Q_1 = 25$, Q_3$ = 75$$, and $$IQR = 50. $$What is the upper fence (UF) for detecting outliers?

Accepted Answer

$$UF = Q_3$ + 1.5 	imes IQR = 75 + 1.5 	imes 50 = 150$$

Fundamental Concepts and Formulas in Statistics

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