Q: A class of 35 students has an average grade of 70. The 20 male students have an average of 73. What is the average grade of the 15 female students?

$\frac{70 \times 35 - 73 \times 20}{15} = 66$

Q: A Poisson process has a mean of 5.1 items sold in 8 hours. What is the mean number of items sold in 1.5 hours?

$\frac{5.1}{8} \times 1.5 = 0.9562$

Question 1

An urn contains 4 white balls, 3 black balls, and 5 red balls. What is the probability of drawing a ball that is neither white nor red?

Accepted Answer

$\frac{3}{12} = 0.25$

Question 2

Two events A and B are said to be mutually exclusive if:

Accepted Answer

They cannot occur at the same time, so $P(A \cap B) = 0$

Question 3

A store sells an average of 5.1 items per day. Assuming the number of items sold follows a Poisson distribution, what is the probability of selling more than 2 items in a day?

Accepted Answer

$1 - (e^{-5.1} + 5.1e^{-5.1} + \frac{5.1^2}{2}e^{-5.1}) = 0.88357$

Question 4

If $X \sim 	ext{Poisson}(\lambda)$, what is the standard deviation of $X$?

Accepted Answer

$\sqrt{\lambda}$

Question 5

A website requires passwords of 5 characters: the first three must be letters (A-Z), the last two must be digits (0-9), and the digits must be distinct. How many possible passwords are there?

Accepted Answer

$26^3 \cdot 10 \cdot 9 = 1581840$

Question 6

In a study, 212 consumers rated four brands of detergent. 30 rated Brand A as 'good' or 'very good'. What percentage of all consumers found Brand A to be 'good' or 'very good'?

Accepted Answer

$\frac{30}{212} = 0.141 = 14.1\%$

Question 7

Given the data set: 12, 13, 13, 21, 22, 23, 25, 26, 31, 34, what is the interquartile range (IQR)?

Accepted Answer

$26 - 13 = 13$

Question 8

The coefficient of variation (CV) is calculated as:

Accepted Answer

$CV = \frac{\sigma}{\mu}$

Question 9

Stock A has a geometric mean return of $1.0020$ and a standard deviation of $0.151$. What is its coefficient of variation (CV)?

Accepted Answer

$\frac{0.151}{1.0020} = 0.151$

Question 10

A sample of 24 workers has evacuation times (in seconds) sorted as follows: 324, 324, 346, 364, 366, 369, 370, 370, 372, 373, 374, 375, 375, 392, 393, 394, 395, 397, 402, 404, 404, 404, 404, 424. What is the median evacuation time?

Accepted Answer

$\frac{366 + 369}{2} = 367.5$

Question 11

If the maximum evacuation time (currently 424 seconds) increases, by how much could it increase without affecting the sample median value?

Accepted Answer

It can increase by any amount without affecting the median

Question 12

A class of 35 students has an average grade of 70. The 20 male students have an average of 73. What is the average grade of the 15 female students?

Accepted Answer

$\frac{70 	imes 35 - 73 	imes 20}{15} = 66$

Question 13

Mavrik can draw from Urn 1 (7 purple, 5 yellow, 3 green balls) or Urn 2 (2 white, 1 black ball). If purple = $2,000, yellow = $3,000, green = $10,000, white = $3,500, black = $5,000, which urn should he choose based on expected value?

Accepted Answer

Urn 2, because $E(X_2) = 4,000$ is higher than $E(X_1) = 3,933.33$

Question 14

For a Poisson random variable with mean $\lambda = 1.275$, what is the probability of selling fewer than 2 items?

Accepted Answer

$2.275 \cdot e^{-1.275} = 0.6357$

Question 15

If two events A and B are independent, which of the following is true?

Accepted Answer

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

Question 16

A Poisson process has a mean of 5.1 items sold in 8 hours. What is the mean number of items sold in 1.5 hours?

Accepted Answer

$\frac{5.1}{8} 	imes 1.5 = 0.9562$

Question 17

Which of the following best describes the geometric mean of $n$ values $x_1, x_2, ..., x_n$?

Accepted Answer

$\left( \prod_{i=1}^n x_i \right)^{1/n}$

Statistics for Business: Key Concepts, Applications, and Problem-Solving Techniques

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