Question 1

Which of the following best describes a uniform probability distribution?

Accepted Answer

A distribution where all outcomes are equally likely within a given interval.

Question 2

Suppose the time it takes for a chemical reaction to complete is uniformly distributed between 10 and 20 minutes. What is the probability that the reaction takes less than 15 minutes?

Accepted Answer

$$P(X \u003c 15) = \frac{15-10}{20-10} = \frac{5}{10} = 0.5$$

Question 3

What is the mean of a uniform distribution defined on the interval $$[a, b]$$?

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

A medical test result is normally distributed with a mean of 100 and a standard deviation of 15. What is the $$z-$$score for a test result of 130?

Accepted Answer

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

Question 6

If $$X is $$uniformly distributed on $$[2, 8]$$, what is the probability density function (pdf) of $$X$$?

Accepted Answer

$$f(x) = \frac{1}{8-2} = \frac{1}{6}$$ for $$2 \leq x \leq 8$$

Question 7

Which real-world scenario is best modeled by a standard normal distribution?

Accepted Answer

Heights of adult humans standardized to mean 0 and standard deviation 1.

Question 8

What is the variance of a uniform distribution on $$[a, b]$$?

Accepted Answer

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

Question 9

A student scores 1.5 standard deviations above the mean on a standardized test. What is their $$z-$$score?

Accepted Answer

$$z = 1.5$$

Question 10

If the probability that a randomly selected patient has a cholesterol level between 180 and 200 mg/dL is 0.25 under a normal distribution, what does this probability represent?

Accepted Answer

The area under the normal curve between $$x = 180$$ and $$x = 200.$$

Question 11

Which formula is used to convert a value $$x$$ from a normal distribution with mean $$\mu$$ and standard deviation $$\sigma to a$$ $$z-$$score?

Accepted Answer

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

Question 12

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

Accepted Answer

It is skewed to the right.

Question 13

A uniform distribution is used to model the waiting time for a bus that arrives randomly between 8:00 and 8:30 AM. What is the probability that the bus arrives before 8:10 AM?

Accepted Answer

$$P(X \u003c 10) = \frac{10}{30} = \frac{1}{3}$$

Question 14

If a variable $$X$$ follows a standard normal distribution, what is $$P(X \u003e 0)$$?

Accepted Answer

$$0.5$$

Question 15

Which of the following best describes the shape of the standard normal distribution?

Accepted Answer

Bell-shaped and symmetric about the mean.

Question 16

A hospital records the time (in minutes) patients wait for a doctor, which is uniformly distributed between 5 and 25 minutes. What is the expected waiting time?

Accepted Answer

$$\frac{5+25}{2} = 15$$

Question 17

Which scenario would NOT be appropriately modeled by a uniform distribution?

Accepted Answer

The distribution of exam scores in a class.

Question 18

If $$X is $$uniformly distributed on $$[0, 12]$$, what is $$P(4 \u003c X \u003c 8)$$?

Accepted Answer

$$P(4 \u003c X \u003c 8) = \frac{8-4}{12-0} = \frac{4}{12} = \frac{1}{3}$$

Question 19

A researcher standardizes test scores to have a mean of 0 and a standard deviation of 1. What distribution do the scores now follow?

Accepted Answer

Standard normal distribution

Study Guidance: Uniform and Standard Normal Distributions

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