Question 1

Suppose the time (in minutes) your friend is late is uniformly distributed between 0 and 30 minutes. What is the probability that your friend is between 5 and 15 minutes late?

Accepted Answer

$$\frac{1}{3}$$

Question 2

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

Accepted Answer

The value of the pdf at a specific point gives the probability that the random variable takes that value.

Question 3

For a continuous random variable $$X$$ with pdf $$f(x)$$, what does the area under the curve between $$a$$ and $$b$$ represent?

Accepted Answer

The probability that $$X$$ takes a value in the interval $$[a, b].$$

Question 4

A pdf for a uniform distribution on the interval $$[0, 30] is $$given by $$f(x) = c$$ for $$0 \leq x \leq 30. $$What is the value of $$c$$?

Accepted Answer

$$c = \frac{1}{30}$$

Question 5

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

Accepted Answer

It has inflection points at $$\mu - 2\sigma$$ and $$\mu + 2\sigma.$$

Question 6

According to the Empirical Rule, approximately what percentage of the data falls within two standard deviations of the mean in a normal distribution?

Accepted Answer

95%

Question 7

If the mean height of three-year-old females is $$\mu = 38.72$$ inches and the standard deviation is $$\sigma = 3.17$$ inches, what is the interval that contains approximately 68% of the heights?

Accepted Answer

$$[35.55, 41.89]$$

Question 8

Which of the following best describes the difference between a uniform and a normal distribution?

Accepted Answer

A uniform distribution has equal probability for all intervals of equal length, while a normal distribution is bell-shaped and symmetric.

Question 9

For a normal distribution, what is the significance of the inflection points?

Accepted Answer

They are the points where the curvature of the graph changes, located at $$\mu - \sigma$$ and $$\mu + \sigma.$$

Question 10

Suppose the serum total cholesterol for males aged 20-29 is normally distributed with $$\mu = 180 mg/dL $$and $$\sigma = 36.2 mg/dL. $$What is the probability that a randomly selected male from this group has cholesterol greater than 200 mg/dL, given that the area under the curve to the right of $$x = 200 is 0.2903$$?

Accepted Answer

0.2903

Question 11

Which of the following is a correct interpretation of the area under the normal curve between $$x = 200$$ and $$x = 220$$ for cholesterol levels?

Accepted Answer

It is the proportion of the population with cholesterol between 200 and 220 mg/dL.

Question 12

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

Accepted Answer

$$P(X = x) = 0$$

Question 13

A continuous random variable $$X$$ has a pdf $$f(x)$$ such that $$f(x) = 2$$ for $$0 \u003c x \u003c 0.5. $$What is the probability that $$X is $$between 0.1 and 0.4?

Accepted Answer

$$0.3$$

Question 14

If a normal distribution has mean $$\mu = 10$$ and standard deviation $$\sigma = 2$$, which interval contains approximately 99.7% of the data?

Accepted Answer

$$[4, 16]$$

Question 15

Which of the following best describes the mean of a density curve?

Accepted Answer

It is the balance point at which the curve would balance if made of solid material.

Question 16

A histogram of test scores is symmetric and bell-shaped. Which model is most appropriate to describe the distribution of scores?

Accepted Answer

Normal distribution

Question 17

If the area under the normal curve to the left of $$x = \mu is 0.5$$, what does this imply about the distribution?

Accepted Answer

The distribution is symmetric about the mean.

Question 18

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

Accepted Answer

The curve has inflection points at $$\mu - \sigma$$ and $$\mu + \sigma.$$

Question 19

A continuous random variable $$X$$ has a pdf $$f(x)$$ such that $$f(x) = k$$ for $$a \u003c x \u003c b. $$What must $$k be in $$terms of $$a$$ and $$b$$?

Accepted Answer

$$k = \frac{1}{b - a}$$

Probability Density Functions, Uniform Distribution, and the Normal Distribution

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