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 is NOT true for a probability density function $$f(x) of a $$continuous random variable?

Accepted Answer

The probability that $$X$$ equals a specific value is positive

Question 3

A company tracks the delivery time (in minutes) for its drivers. If delivery time is modeled as a continuous random variable, which of the following is the best way to interpret $$P[30 \leq X \leq 40]$$?

Accepted Answer

The area under the probability density curve between $$x=30$$ and $$x=40.$$

Question 4

Which of the following is a defining property of the normal distribution?

Accepted Answer

It is symmetric with respect to the mean $$\mu.$$

Question 5

If $$X \sim N(\mu, \$$sigma^2$$)$$, which formula correctly standardizes $$X to a $$standard normal variable $$Z$$?

Accepted Answer

$$Z = \frac{X - \mu}{\sigma}$$

Question 6

Suppose the mass of lion cubs is normally distributed with mean $$\mu = 8 kg $$and standard deviation $$\sigma = 1.5 kg. $$What is the $$z-$$score for a cub weighing 10 kg?

Accepted Answer

$$z = \frac{10 - 8}{1.5} = 1.33$$

Question 7

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

Accepted Answer

It is skewed to the left.

Question 8

If $$P(Z \leq 1) = 0.8413$$ for a standard normal variable $$Z$$, what is $$P(Z \u003e 1)$$?

Accepted Answer

$$0.1587$$

Question 9

A marketing manager wants to find the minimum purchase value $$x$$ for the top 15% of customers, given that purchase values are normally distributed with mean $$\mu = 500$$ and $$\sigma = 100. $$Which is the correct sequence of steps?

Accepted Answer

Find the $$z-$$value for the 85th percentile, then use $$x = \mu + z\sigma.$$

Question 10

Which of the following best explains why the area under a probability density function must equal 1?

Accepted Answer

Because the total probability for all possible outcomes must be 1.

Question 11

If $$Z is a $$standard normal variable, which of the following is true due to symmetry?

Accepted Answer

$$P(Z \leq -a) = 1 - P(Z \leq a)$$

Question 12

A delivery company finds that the daily distance traveled by its vehicles is normally distributed with mean $$\mu = 120 km $$and standard deviation $$\sigma = 15 km. $$What is the probability that a randomly selected vehicle travels between 105 km and 135 km in a day?

Accepted Answer

First, calculate $$z-$$scores: $$z_1$$ = \frac{105-120}{15} = -1$$, z_2$ = \frac{135-120}{15} = 1. $$Then, $$P(-1 \leq Z \leq 1) = 0.6826.$$

Question 13

Which of the following is NOT a step in finding an $$x-$$value for a given probability in a normal distribution?

Accepted Answer

Multiply the probability by the mean.

Question 14

If $$X is a $$continuous random variable with probability density function $$f(x)$$, what is the probability that $$X$$ takes on any single specific value $$c$$?

Accepted Answer

$$P(X = c) = 0$$

Question 15

Which of the following best describes the effect of increasing the standard deviation $$\sigma in a $$normal distribution $$N(\mu, \$$sigma^2$$)$$?

Accepted Answer

The curve becomes wider and shorter.

Question 16

A random variable $$X is $$normally distributed with mean $$\mu = 50$$ and standard deviation $$\sigma = 10. $$What $$x-$$value corresponds to the 97.5th percentile?

Accepted Answer

Find $$z = 1.96$$ from the standard normal table, then $$x = 50 + 1.96 	imes 10 = 69.6$$

Probability Distributions of Continuous Random Variables and the Normal Distribution

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