Question 1

Which of the following best describes the shape and symmetry of a normal distribution?

Accepted Answer

It is bell-shaped and symmetric.

Question 2

Given a continuous random variable with a uniform distribution over the interval [a, b], what is the formula for the probability that the variable falls between $$c$$ and $$d$$ ($$a \leq c \u003c d \leq b$$)?

Accepted Answer

$$\frac{d-c}{b-a}$$

Question 3

What is the total area under the curve of any continuous probability distribution?

Accepted Answer

It is equal to 1.

Question 4

Which formula is used to convert a value $$x$$ from a nonstandard normal distribution to a $$z-$$score in the standard normal distribution?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 5

A hospital records the waiting times for patients in the emergency room. If the waiting times are uniformly distributed between 0 and 10 minutes, what is the probability that a randomly selected patient waits at least 2 minutes?

Accepted Answer

$$\frac{8}{10}$$

Question 6

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

Accepted Answer

The mean is 0 and the standard deviation is 1.

Question 7

If a normal distribution has a mean of $$\mu = 50$$ and a standard deviation of $$\sigma = 5$$, what is the $$z-$$score for $$x = 60$$?

Accepted Answer

$$z = 2$$

Question 8

Which property is shared by all normal distributions regardless of their mean and standard deviation?

Accepted Answer

They are bell-shaped and symmetric.

Question 9

A continuous random variable $$X is $$uniformly distributed between 5 and 15. What is the probability that $$X is $$less than 10?

Accepted Answer

$$\frac{10-5}{15-5} = \frac{5}{10} = 0.5$$

Question 10

In a normal distribution, what percentage of values fall within one standard deviation of the mean?

Accepted Answer

Approximately 68%

Question 11

A pharmaceutical company measures the time it takes for a drug to take effect, which follows a normal distribution with mean $$\mu = 30$$ minutes and standard deviation $$\sigma = 4$$ minutes. What is the $$z-$$score for a patient whose drug takes effect in 38 minutes?

Accepted Answer

$$z = 2$$

Question 12

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

Accepted Answer

The curve is rectangular.

Question 13

If the waiting time for a bus is uniformly distributed between 0 and 20 minutes, what is the probability that a passenger waits between 5 and 15 minutes?

Accepted Answer

$$\frac{15-5}{20-0} = \frac{10}{20} = 0.5$$

Question 14

A normal distribution has a mean of $$\mu = 100$$ and a standard deviation of $$\sigma = 10. $$What value of $$x$$ corresponds to a $$z-$$score of $$-1.5$$?

Accepted Answer

$$x = 85$$

Question 15

Which of the following real-world scenarios is most likely to follow a normal distribution?

Accepted Answer

Heights of adult humans in a population.

Question 16

If a continuous random variable has a uniform distribution, what is the shape of its probability density function?

Accepted Answer

Rectangular

Question 17

A normal distribution is defined by which two parameters?

Accepted Answer

Mean and standard deviation

Question 18

If the area under the curve of a probability distribution is not equal to 1, what does this indicate?

Accepted Answer

The distribution is not a valid probability distribution.

Question 19

A student scores 80 on a test where the mean is 70 and the standard deviation is 5. What is the student's $$z-$$score?

Accepted Answer

$$z = 2$$

Question 20

Which of the following best explains why probabilities in continuous distributions are found using areas under the curve?

Accepted Answer

Because the area under the curve represents the likelihood of outcomes within a range.

Study Notes: Normal Probability Distributions

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