Q: A business wants to know the 90th percentile of a normally distributed sales variable with $$\mu = 200$$ and $$\sigma = 20. $$The $$z-$$value for the 90th percentile is approximately 1.28. What is the corresponding $$x$$ value?

$$x = 200 + 1.28 \times 20 = 225.6$$

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

Which property must every probability density function $$f(x)$$ satisfy?

Accepted Answer

$$\int_{-\infty}^{\infty} f(x) dx = 1$$

Question 3

The area under the probability density curve between $$a$$ and $$b$$ represents:

Accepted Answer

The probability that $$X$$ lies in the interval $$[a, b]$$

Question 4

A normal distribution with mean $$\mu$$ and variance $$\$$sigma^2$$ is denoted as:

Accepted Answer

$$N(\mu, \$$sigma^2$$)$$

Question 5

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

Accepted Answer

It is skewed to the right.

Question 6

If $$X \sim N(100, 16)$$, what is the standard deviation of $$X$$?

Accepted Answer

$$4$$

Question 7

The standard normal distribution has which of the following parameters?

Accepted Answer

$$\mu = 0, \sigma = 1$$

Question 8

What is the $$z-$$score formula for converting an $$x$$ value from a normal distribution $$N(\mu, \$$sigma^2$$) to $$the standard normal distribution?

Accepted Answer

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

Question 9

Suppose the delivery time for a courier company is normally distributed with mean $$\mu = 30$$ minutes and standard deviation $$\sigma = 5$$ minutes. What is the $$z-$$score for a delivery that takes 40 minutes?

Accepted Answer

$$z = 2$$

Question 10

If $$Z \sim N(0,1)$$, what is $$P(Z \leq 0)$$?

Accepted Answer

$$0.5$$

Question 11

Which of the following best describes the effect of increasing the standard deviation $$\sigma of a $$normal distribution while keeping the mean $$\mu$$ constant?

Accepted Answer

The curve becomes shorter and wider.

Question 12

A company wants to find the minimum purchase value $$x$$ for the top 10% of customers. Which steps should they follow?

Accepted Answer

Find the $$z-$$value for the top 10%, then convert it to $$x$$ using $$x = \mu + z\sigma.$$

Question 13

If $$P(Z \leq 1.5) = 0.9332$$, what is $$P(Z \u003e 1.5)$$?

Accepted Answer

$$0.0668$$

Question 14

Which of the following is a correct interpretation of $$P(a \leq Z \leq b)$$ for the standard normal distribution?

Accepted Answer

It is the area under the curve between $$a$$ and $$b.$$

Question 15

A histogram of lion cub masses is constructed from a sample of 100 cubs. As the sample size increases to 5000, the histogram will:

Accepted Answer

Approach the shape of the underlying probability density curve.

Question 16

If $$P(Z \leq a) = 0.8$$, what is $$P(Z \u003e a)$$?

Accepted Answer

$$0.2$$

Question 17

Which of the following is true about the probability of a continuous random variable taking any exact value $$x$$?

Accepted Answer

It is always 0.

Question 18

Given $$X \sim N(50, 25)$$, what is the probability that $$X is $$less than 50?

Accepted Answer

$$0.5$$

Question 19

If $$P(Z \leq 2) = 0.9772$$ and $$P(Z \leq 1) = 0.8413$$, what is $$P(1 \leq Z \leq 2)$$?

Accepted Answer

$$0.1359$$

Question 20

A business wants to know the 90th percentile of a normally distributed sales variable with $$\mu = 200$$ and $$\sigma = 20. $$The $$z-$$value for the 90th percentile is approximately 1.28. What is the corresponding $$x$$ value?

Accepted Answer

$$x = 200 + 1.28 	imes 20 = 225.6$$

Probability Distributions of Continuous Random Variables and the Normal Distribution

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