Question 1

Which of the following statements about continuous random variables is TRUE?

Accepted Answer

The probability that a continuous random variable falls within a specific interval is given by the area under its probability density function over that interval.

Question 2

Suppose the time (in minutes) it takes for an Amazon package to arrive after 10:00 am is uniformly distributed between 0 and 60 minutes. What is the probability that the package arrives between 10:15 am and 10:30 am?

Accepted Answer

$$P = \frac{15}{60} = 0.25$$

Question 3

Which of the following is NOT a property of a probability density function (pdf) for a continuous random variable?

Accepted Answer

The sum of the pdf values at all possible points equals 1.

Question 4

A continuous random variable $$X is $$uniformly distributed on the interval $$[0, 60]. $$What is the value of its probability density function $$f(x)$$ for $$0 \leq x \leq 60$$?

Accepted Answer

$$f(x) = \frac{1}{60}$$

Question 5

Which of the following best describes the shape and properties of a normal distribution?

Accepted Answer

It is symmetric about its mean, and the mean, median, and mode are all equal.

Question 6

The Empirical Rule states that for a normal distribution, approximately what percentage of data falls within two standard deviations of the mean?

Accepted Answer

95%

Question 7

If the total cholesterol levels for males aged 20-29 are normally distributed with mean $$\mu = 180 mg/dL $$and standard deviation $$\sigma = 36.2 mg/dL$$, what is the $$z-$$score for a cholesterol level of 200 mg/dL?

Accepted Answer

$$z = \frac{200 - 180}{36.2} \approx 0.55$$

Question 8

For a standard normal distribution, what is the area to the left of $$z = 1.33$$?

Accepted Answer

$$0.9082$$

Question 9

Which of the following statements about the area under the normal curve is correct?

Accepted Answer

The total area under the normal curve is always equal to 1.

Question 10

A pediatrician finds that the heights of three-year-old girls are normally distributed with mean $$\mu = 38.72$$ inches and standard deviation $$\sigma = 3.17$$ inches. What is the height corresponding to the 20th percentile?

Accepted Answer

$$x = 38.72 + (-0.84)(3.17) \approx 36.05$$ inches

Question 11

If $$X is a $$normal random variable with mean $$\mu$$ and standard deviation $$\sigma$$, which formula is used to standardize $$X to a$$ $$z-$$score?

Accepted Answer

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

Question 12

A student scores 120 on an IQ test that is normally distributed with mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the $$z-$$score for this student?

Accepted Answer

$$z = \frac{120 - 100}{15} = 1.33$$

Question 13

Which of the following is a correct interpretation of the area under the normal curve to the right of $$x = 200 mg/dL $$for cholesterol levels in young males?

Accepted Answer

It represents the proportion of young males with cholesterol greater than 200 mg/dL.

Question 14

If the area to the left of $$z = 1.33 is$$ $$0.9082$$, what is the area to the right of $$z = 1.33$$?

Accepted Answer

$$1 - 0.9082 = 0.0918$$

Question 15

The mathematics portion of the SAT is approximately normally distributed with mean $$\mu = 516$$ and standard deviation $$\sigma = 116. $$What SAT score corresponds to the 95th percentile?

Accepted Answer

$$x = 516 + (1.645)(116) \approx 707$$

Question 16

Which of the following is TRUE about the tails of the normal distribution?

Accepted Answer

The tails approach, but never touch, the horizontal axis.

Question 17

A continuous random variable $$X$$ has a uniform distribution on $$[0, 60]. If $$there is a 30% probability that your package arrives within $$t$$ minutes after 10 am, what is the value of $$t$$?

Accepted Answer

$$t = 0.3 	imes 60 = 18$$ minutes

Question 18

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

Accepted Answer

The area under the curve is less than 1.

Question 19

If $$X is a $$continuous random variable, which of the following statements is correct?

Accepted Answer

$$P(a \u003c X \u003c b)$$ can be found by integrating the pdf from $$a to$$ $$b$$

Continuous and Normal Probability Distributions

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