Question 1

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

Accepted Answer

It can take on any value within a specified interval.

Question 2

In a normal probability distribution, what is the relationship between the mean and the median?

Accepted Answer

The mean and median are equal.

Question 3

Suppose the time customers spend on the phone follows a normal distribution with a mean of 12 minutes and a standard deviation of 3 minutes. What is the z-score for a customer who spends 14 minutes on the phone?

Accepted Answer

$$z = \frac{14 - 12}{3} = 0.67$$

Question 4

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

Accepted Answer

It is skewed to the right.

Question 5

If the annual snowfall in a city is normally distributed with a mean of 46 inches and a standard deviation of 5 inches, what is the probability that snowfall will be more than 56 inches next year? (Assume $$z = 2$$ for 56 inches)

Accepted Answer

$$P(z \u003e 2) = 1 - 0.9772 = 0.0228$$

Question 6

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

Accepted Answer

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

Question 7

The exponential probability distribution is commonly used to model which type of scenario?

Accepted Answer

The time between arrivals of customers at a store.

Question 8

Which of the following is the formula for the exponential probability density function?

Accepted Answer

$$f(x) = \lambda e$$^{-\lambda x}$$

Question 9

A company expects that 95% of its customers will require less than 18 minutes per call. If call times are normally distributed with mean 12 and standard deviation 3, what is the $$z-$$score that corresponds to the 95th percentile?

Accepted Answer

$$z = 1.645$$

Question 10

Which of the following statements about the exponential distribution is TRUE?

Accepted Answer

It is right-skewed and described by only one parameter.

Question 11

Suppose the average time between customer arrivals at a counter is 4 minutes. What is the probability that the next customer will arrive within 2 minutes? (Use $$P(x \u003c a) = 1 - e$$^{-\lambda a}$$, $$\lambda = 0.25$$)

Accepted Answer

$$P(x \u003c 2) = 1 - e$$^{-0.5}$$ \approx 0.3935$$

Question 12

When can the normal distribution be used to approximate the binomial distribution?

Accepted Answer

When $$np \geq 5$$ and $$n(1-p) \geq 5$$

Question 13

What is the purpose of the continuity correction when using the normal approximation for the binomial distribution?

Accepted Answer

To adjust for the difference between discrete and continuous variables by adding or subtracting 0.5.

Question 14

If $$X is $$exponentially distributed with mean $$\mu$$, what is the standard deviation of $$X$$?

Accepted Answer

$$\sigma = \mu$$

Question 15

Which Excel function would you use to find the cumulative probability for an exponential distribution?

Accepted Answer

=EXPON.DIST(x, lambda, TRUE)

Question 16

A group of workers has incomes normally distributed with $$\mu = 45,000$$ and $$\sigma = 5,000. $$What is the probability that a randomly selected worker earns between $$38,000$$ and $$48,000$$? (Use $$z_1 = -1.4$, z_2$ = 0.6$$; $$P(z_1$) = 0.0808$$, $$P(z_2$) = 0.7257)$$

Accepted Answer

$$P = P(z_2$) - P(z_1$) = 0.7257 - 0.0808 = 0.6449$$

Question 17

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

Accepted Answer

It is right-skewed.

Question 18

If a normal distribution has mean $$\mu$$ and standard deviation $$\sigma$$, what effect does increasing $$\sigma$$ have on the shape of the distribution?

Accepted Answer

It increases the spread, making the curve flatter and wider.

Chapter 6: Continuous Probability Distributions – Study Notes for Business Statistics

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