Question 1

Which of the following is a key difference between a discrete random variable and a continuous random variable?

Accepted Answer

A discrete random variable can only take on specific, countable values, while a continuous random variable can take on any value within a given range.

Question 2

Suppose you are given the following probability distribution for a random variable $$X$$: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3. $$What is the mean (expected value) of $$X$$?

Accepted Answer

$$E(X) = 2.1$$

Question 3

Which of the following conditions must be satisfied for a probability distribution to be valid?

Accepted Answer

The sum of all probabilities must equal 1.

Question 4

A hospital records the number of patients admitted each day. Which type of random variable best describes this scenario?

Accepted Answer

Discrete random variable

Question 5

Given a discrete probability distribution with $$E(X) = 4$$ and $$Var(X) = 9$$, what is the standard deviation of $$X$$?

Accepted Answer

$$\sqrt{9} = 3$$

Question 6

Which of the following statements about the normal distribution is TRUE?

Accepted Answer

The mean, median, and mode are all equal.

Question 7

If a variable $$X is $$normally distributed with mean $$\mu = 50$$ and standard deviation $$\sigma = 10$$, what is the probability that $$X \u003c 60$$? (Use the standard normal table and round your answer to two decimal places.)

Accepted Answer

0.84

Question 8

Which Excel function would you use to find the cumulative probability that a normally distributed variable is less than a given value $$x$$?

Accepted Answer

NORM.DIST

Question 9

A researcher wants to find the value $$x$$ such that the probability of a normally distributed variable being less than $$x is 0.95. $$Which Excel function should they use?

Accepted Answer

NORM.INV

Question 10

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

Accepted Answer

It becomes approximately normal, regardless of the shape of the population distribution.

Question 11

A population has mean $$\mu = 100$$ and standard deviation $$\sigma = 20. If a $$sample of size $$n = 25 is $$taken, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\frac{20}{\sqrt{25}} = 4$$

Question 12

Which of the following best describes a point estimate in the context of confidence intervals?

Accepted Answer

A single value used to estimate a population parameter.

Question 13

A 95% confidence interval for the mean is calculated as $$\bar{x} \pm E. $$What does $$E$$ represent?

Accepted Answer

The margin of error

Question 14

A pharmaceutical company wants to estimate the proportion of patients who experience side effects from a new drug. Which type of confidence interval should they construct?

Accepted Answer

Confidence interval for population proportion

Question 15

If the population standard deviation $$\sigma is $$known, which formula is used to calculate the margin of error $$E$$ for a confidence interval for the mean?

Accepted Answer

$$E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$

Question 16

A sample of 100 students has a mean test score of 80 and a known population standard deviation of 10. What is the 95% confidence interval for the population mean? (Use $$z_{0.025} = 1.96)$$

Accepted Answer

$$80 \pm 1.96 \cdot \frac{10}{\sqrt{100}} = 80 \pm 1.96$$

Question 17

Which of the following is NOT a property of the sampling distribution of the sample mean?

Accepted Answer

It is always normally distributed, regardless of sample size.

Question 18

A survey finds that 40% of respondents prefer online shopping. If the sample size is 200, what is the point estimate for the population proportion?

Accepted Answer

$$\hat{p} = 0.40$$

Question 19

Which scenario best illustrates the application of the Central Limit Theorem?

Accepted Answer

A researcher averages the heights of 50 randomly selected adults to estimate the average height in a city.

Question 20

A discrete probability distribution is given by $$P(X=0)=0.1$$, $$P(X=1)=0.3$$, $$P(X=2)=0.6. $$What is the variance of $$X$$?

Accepted Answer

$$Var(X) = 0.49$$

Study Notes: Discrete and Normal Probability Distributions, Sampling, and Confidence Intervals

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