Q: Which of the following is true about the cumulative distribution function (CDF) $$F(x) of a $$continuous random variable?

$$F(x) is $$non-decreasing and $$\lim_{x \to \infty} F(x) = 1$$

Question 1

Which of the following best describes the difference between a continuous and a discrete random variable?

Accepted Answer

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

Question 2

Which of the following is a property of the normal distribution?

Accepted Answer

It is symmetric about the mean.

Question 3

Given a standard normal distribution, what is $$P(Z \u003c 0)$$?

Accepted Answer

$$0.5$$

Question 4

Which of the following statements about probability density functions (PDFs) is correct?

Accepted Answer

The area under the PDF curve over all possible values equals 1.

Question 5

If $$Z is a $$standard normal random variable, what is $$P(Z \u003e 1.96)$$?

Accepted Answer

$$0.025$$

Question 6

Which of the following best defines a sampling distribution?

Accepted Answer

The probability distribution of a statistic based on a random sample.

Question 7

Suppose the heights of adult males are normally distributed with mean $$\mu = 175 cm $$and standard deviation $$\sigma = 10 cm. $$What is the $$z-$$score for a man who is 190 cm tall?

Accepted Answer

$$z = 1.5$$

Question 8

Which of the following is NOT a property of the normal distribution?

Accepted Answer

It is bimodal.

Question 9

If $$X is a $$continuous random variable with probability density function $$f(x)$$, which of the following represents the probability that $$X$$ falls between $$a$$ and $$b$$?

Accepted Answer

$$\int_a^b f(x) dx$$

Question 10

A factory produces light bulbs with lifetimes that are normally distributed with mean $$\mu = 1200$$ hours and standard deviation $$\sigma = 100$$ hours. What is the probability that a randomly selected bulb lasts more than 1400 hours?

Accepted Answer

$$P(Z \u003e 2) = 0.0228$$

Question 11

Which of the following is true about the cumulative distribution function (CDF) $$F(x) of a $$continuous random variable?

Accepted Answer

$$F(x) is $$non-decreasing and $$\lim_{x 	o \infty} F(x) = 1$$

Question 12

If $$Z is a $$standard normal random variable, what is $$P(-1 \u003c Z \u003c 1)$$?

Accepted Answer

$$0.6826$$

Question 13

Which of the following best describes the probability mass function (PMF)?

Accepted Answer

It gives the probability that a discrete random variable is exactly equal to some value.

Question 14

Suppose $$Z is a $$standard normal random variable. What is $$P(Z \u003c -2.33)$$?

Accepted Answer

$$0.0099$$

Question 15

A researcher wants to find the probability that a randomly selected value from a normal distribution is between the mean and one standard deviation above the mean. What is this probability?

Accepted Answer

$$0.3413$$

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