Question 1

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

Accepted Answer

The mean, median, and mode are all equal to 0.

Question 2

Suppose the length of a class is uniformly distributed between 50.3 and 50.9 minutes. What is the probability that a randomly selected class lasts between 50.3 and 50.6 minutes?

Accepted Answer

$$P(50.3 \u003c x \u003c 50.6) = \frac{50.6 - 50.3}{50.9 - 50.3} = \frac{0.3}{0.6} = 0.5$$

Question 3

Given a normal distribution with mean $$\mu = 100$$ and standard deviation $$\sigma = 15$$, what is the $$z-$$score for $$x = 130$$?

Accepted Answer

$$z = \frac{130 - 100}{15} = 2$$

Question 4

Which formula represents the probability density function (pdf) for a normal random variable $$x$$ with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

$$f(x) = \frac{1}{\sqrt{2\pi\$$sigma^2$$}} e$$^{-\frac{(x-\mu)^2}$${2\$$sigma^2$$}}$$

Question 5

If $$z is a $$standard normal random variable, what is $$P(z \u003c 1.33)$$ according to the standard normal table?

Accepted Answer

$$P(z \u003c 1.33) = 0.9082$$

Question 6

Which of the following statements about the area under the standard normal curve is TRUE?

Accepted Answer

The area between $$z = -1$$ and $$z = 1 is $$approximately 68%.

Question 7

A continuous random variable $$X$$ has a uniform distribution on the interval $$[a, b]. $$What is the mean $$\mu_X of$$ $$X$$?

Accepted Answer

$$\mu_X = \frac{a + b}{2}$$

Question 8

Which of the following best describes a probability density function (pdf) for a continuous random variable?

Accepted Answer

The area under the curve between two values gives the probability that the variable falls within that interval.

Question 9

If $$X is a $$normal random variable with $$\mu = 50$$ and $$\sigma = 10$$, what is the probability that $$X is $$less than 65? (Use the standard normal table.)

Accepted Answer

$$z = \frac{65 - 50}{10} = 1.5$$, so $$P(X \u003c 65) = P(z \u003c 1.5) = 0.9332$$

Question 10

Which of the following is a correct interpretation of the Empirical Rule for a normal distribution?

Accepted Answer

About 95% of the data falls within two standard deviations of the mean.

Question 11

A random variable $$X is $$said to be continuous if:

Accepted Answer

It can take on any value within an interval.

Question 12

For a standard normal distribution, what is the probability that $$z is $$between -1 and 1?

Accepted Answer

$$P(-1 \u003c z \u003c 1) \approx 0.68$$

Question 13

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

Accepted Answer

It is skewed to the right.

Question 14

If $$X is a $$continuous random variable with a uniform distribution on $$[a, b]$$, what is the variance $$\$$sigma^2$$ of X$?

Accepted Answer

$$\$$sigma^2$$ = \frac{(b-a)^2$}{12}$$

Question 15

A probability distribution is called a probability density function (pdf) if:

Accepted Answer

The total area under the curve is 1.

Question 16

If $$z is a $$standard normal random variable, what is $$P(z \u003e 1.55)$$?

Accepted Answer

$$P(z \u003e 1.55) = 1 - P(z \u003c 1.55) = 1 - 0.9394 = 0.0606$$

Question 17

Which of the following best describes the relationship between the mean, median, and mode in a normal distribution?

Accepted Answer

They are all equal.

Question 18

A random variable $$Z$$ has a standard normal distribution. What is the probability that $$Z is $$less than -0.5?

Accepted Answer

$$P(Z \u003c -0.5) = 0.3085$$

Question 19

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

Accepted Answer

It has a mean of 0 and a standard deviation of 1.

Chapter 6: Normal Probability Distributions – Study Notes

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