Question 1

Suppose a delivery is scheduled to arrive at a random time between 10 AM and 11 AM, with all times equally likely. What is the probability that the delivery arrives between 10:15 AM and 10:30 AM?

Accepted Answer

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

Question 2

For a continuous uniform distribution on the interval $$[a, b]$$, what is the height of the probability density function?

Accepted Answer

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

Question 3

Which of the following statements about the area under a probability density function is correct?

Accepted Answer

The total area under the curve is always 1.

Question 4

A normal curve is symmetric about which value?

Accepted Answer

The mean ($$\mu)$$

Question 5

What happens to the shape of a normal curve as the standard deviation ($$\sigma) $$increases, while the mean ($$\mu) $$remains constant?

Accepted Answer

The curve becomes wider and flatter.

Question 6

For a normal distribution, what is the value of $$x at $$the inflection points?

Accepted Answer

$$x = \mu \pm \sigma$$

Question 7

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

Accepted Answer

The curve is skewed to the right.

Question 8

If a random variable $$X is $$uniformly distributed on $$[a, b]$$, what is the probability that $$X$$ falls between $$c$$ and $$d$$ where $$a \leq c \u003c d \leq b$$?

Accepted Answer

$$P = \frac{d-c}{b-a}$$

Question 9

A cholesterol level for males aged 20 to 29 is approximately normally distributed with mean $$\mu = 174$$ and standard deviation $$\sigma = 37. $$What is the $$z-$$score for a cholesterol level of 200?

Accepted Answer

$$z = \frac{200 - 174}{37} \approx 0.70$$

Question 10

Which of the following best describes the role of area under the normal curve in probability?

Accepted Answer

The area under the curve between two values represents the probability that the variable falls within that interval.

Question 11

If the mean of a normal distribution increases while the standard deviation remains constant, what happens to the graph?

Accepted Answer

The graph shifts to the right.

Question 12

Which of the following is a correct interpretation of a probability density function (pdf) for a continuous random variable?

Accepted Answer

The probability that the variable falls within an interval is the area under the pdf over that interval.

Question 13

Suppose the area under the normal curve to the right of $$z = 2.00 is$$ $$0.0228. $$What does this area represent?

Accepted Answer

The probability that a randomly selected value is greater than 2 standard deviations above the mean.

Question 14

Which of the following is true about the mean, median, and mode of a normal distribution?

Accepted Answer

They are all equal.

Question 15

A continuous random variable $$X is $$said to be normally distributed if:

Accepted Answer

Its probability density function is bell-shaped and symmetric about the mean.

Question 16

If the possible values of a uniform density function go from $$a to$$ $$b$$, what is the height of the rectangle?

Accepted Answer

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

Question 17

Which of the following best describes what it means for a continuous variable to be normally distributed?

Accepted Answer

The variable's values are distributed in a bell-shaped, symmetric pattern centered at the mean.

Section 7.1: Properties of the Normal Distribution – Study Notes

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