Q: Suppose the heights of adult women are normally distributed with mean $$\mu = 160 cm $$and standard deviation $$\sigma = 7 cm. $$What is the height $$x$$ such that only 5% of women are shorter than $$x$$?

$$x = 160 + (-1.64) \times 7 = 148.52 cm$$

Question 1

Which of the following statements best describes a continuous random variable?

Accepted Answer

It can take on any value within a given interval.

Question 2

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

Accepted Answer

$$P(a \u003c X \u003c b) = \int_{a}^{b} f(x) \, dx$$

Question 3

A call center’s response time is uniformly distributed between 2 and 12 seconds. What is the probability that a call is answered in less than 5 seconds?

Accepted Answer

$$P(X \u003c 5) = \frac{5-2}{12-2} = 0.3$$

Question 4

Which of the following is a requirement for a function to be a valid probability density function (PDF)?

Accepted Answer

The total area under the curve must be equal to 1.

Question 5

A commuter train arrives every 30 minutes. If a passenger arrives at a random time, what is the probability they wait less than 10 minutes?

Accepted Answer

$$P(X \u003c 10) = \frac{10}{30} = 0.333$$

Question 6

Which of the following best describes the standard normal distribution?

Accepted Answer

A normal distribution with mean $$\mu = 0$$ and standard deviation $$\sigma = 1.$$

Question 7

If $$P(Z \u003c z) = 0.6331$$ for a standard normal variable $$Z$$, what is the value of $$z$$?

Accepted Answer

$$z = 0.34$$

Question 8

Given a standard normal variable $$Z$$, what is $$P(Z \u003e 2.27)$$?

Accepted Answer

$$0.0116$$

Question 9

Suppose the heights of adult women are normally distributed with mean $$\mu = 160 cm $$and standard deviation $$\sigma = 7 cm. $$What is the height $$x$$ such that only 5% of women are shorter than $$x$$?

Accepted Answer

$$x = 160 + (-1.64) 	imes 7 = 148.52 cm$$

Question 10

A bus arrives on average 12 minutes late with a standard deviation of 3 minutes. What is the probability the bus arrives less than 10 minutes late?

Accepted Answer

$$P(X \u003c 10) = P\left(Z \u003c \frac{10-12}{3}\right) = P(Z \u003c -0.67) = 0.2514$$

Question 11

Which calculator function is used to find the probability between two z-scores on a TI-84 calculator?

Accepted Answer

normalcdf

Question 12

If $$P(Z \u003e z) = 0.3409$$ for a standard normal variable $$Z$$, what is the value of $$z$$?

Accepted Answer

$$z = 0.40$$

Question 13

For a non-standard normal variable $$X$$ with mean $$\mu$$ and standard deviation $$\sigma$$, what is the formula for the z-score of a value $$x$$?

Accepted Answer

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

Question 14

If the area to the left of a z-score is 0.9357, what is the z-score?

Accepted Answer

$$z = 1.50$$

Question 15

A probability density function $$f(x) is $$defined on $$[2, 8]$$ and is constant. What is the value of $$f(x)$$?

Accepted Answer

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

Question 16

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

Accepted Answer

The total area under the curve is less than 1.

Question 17

If $$P(Z \u003c z) = 0.8051$$, what is the value of $$z$$?

Accepted Answer

$$z = 0.86$$

Question 18

A normal distribution has mean $$\mu = 20$$ and standard deviation $$\sigma = 5. $$What value $$x$$ corresponds to the 95th percentile?

Accepted Answer

$$x = 20 + 1.64 	imes 5 = 28.2$$

Normal Probability Distributions: Study Notes

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