Q: A fair six-sided die is rolled. What is the probability of rolling a number greater than 4 or an even number?

$$P(\u003e4 \cup \text{even}) = P(\u003e4) + P(\text{even}) - P(\u003e4 \cap \text{even}) = \frac{2}{6} + \frac{3}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$$

Q: A bag contains 5 red, 3 blue, and 2 green marbles. If two marbles are drawn without replacement, what is the probability that both are red?

$$\frac{5}{10} \times \frac{4}{9} = \frac{20}{90} = \frac{2}{9}$$

Question 1

Which of the following best describes the difference between a frequency distribution and a relative frequency distribution?

Accepted Answer

A frequency distribution shows the number of occurrences for each category, while a relative frequency distribution shows the proportion or percentage of occurrences for each category.

Question 2

A hospital records the number of patients admitted each day for a month. Which graphical method is most appropriate to display the distribution of daily admissions?

Accepted Answer

Histogram

Question 3

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

Accepted Answer

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

Question 4

A data set has a mean of 50 and a standard deviation of 5. What is the z-score for a value of 60?

Accepted Answer

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

Question 5

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 6

Which of the following is NOT part of the five-number summary?

Accepted Answer

Mean

Question 7

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

Accepted Answer

Outliers and spread of the data

Question 8

If the probability of rain on any given day is 0.2, what is the probability that it does NOT rain on a randomly selected day?

Accepted Answer

$$1 - 0.2 = 0.8$$

Question 9

A fair six-sided die is rolled. What is the probability of rolling a number greater than 4 or an even number?

Accepted Answer

$$P(\u003e4 \cup 	ext{even}) = P(\u003e4) + P(	ext{even}) - P(\u003e4 \cap 	ext{even}) = \frac{2}{6} + \frac{3}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$$

Question 10

A bag contains 5 red, 3 blue, and 2 green marbles. If two marbles are drawn without replacement, what is the probability that both are red?

Accepted Answer

$$\frac{5}{10} 	imes \frac{4}{9} = \frac{20}{90} = \frac{2}{9}$$

Question 11

Which of the following is a requirement for a probability distribution?

Accepted Answer

The sum of all probabilities must equal 1.

Question 12

A binomial experiment consists of 8 trials with a probability of success $$p = 0.3. $$What is the probability of exactly 2 successes? (Use the binomial formula.)

Accepted Answer

$$P(X=2) = {8 \choose 2} (0.3)^2$ (0.7)^6$

Question 13

A normal distribution has a mean of 100 and a standard deviation of 15. What is the probability that a randomly selected value is between 85 and 115?

Accepted Answer

About 68%, since 85 and 115 are one standard deviation from the mean.

Question 14

According to the Central Limit Theorem, what happens to the sampling distribution of the sample mean as the sample size increases?

Accepted Answer

It approaches a normal distribution, regardless of the shape of the population distribution.

Question 15

A 95% confidence interval for a population mean is calculated as (45, 55). Which of the following is a correct interpretation?

Accepted Answer

We are 95% confident that the true population mean lies between 45 and 55.

Question 16

Which of the following is the correct null hypothesis for testing whether the proportion of defective products in a factory is 5%?

Accepted Answer

$$H_0$$: p = 0.05$$

Question 17

A researcher wants to test if a new medication changes the average recovery time from a disease. Which type of hypothesis test should be used?

Accepted Answer

A hypothesis test for the mean ($$\mu)$$

Question 18

If the correlation coefficient $$r$$ between two variables is 0.85, what does this indicate?

Accepted Answer

There is a strong positive linear relationship between the variables.

Question 19

Given the regression equation $$\hat{y} = 2 + 3x$$, what is the predicted value of $$y$$ when $$x = 4$$?

Accepted Answer

$$\hat{y} = 2 + 3 	imes 4 = 14$$

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