Question

Class is over! Mr. Shrager does not always let his statistics class out on

time. In fact, he seems to end class according to his own "internal clock."
The density curve here models the distribution of Y, the amount of time
after class ends (in minutes) when Mr. Shrager dismisses the class on a
randomly selected day. (A negative value indicates he ended class early.)
A horizontal line is drawn above the axis. A vertical line is drawn from 0
that intersects the horizontal line. The region below the horizontal line
between negative 1 and 4 is shaded.
a. Find and interpret P (Y > 3.75).
b. What is Hy? Explain your answer.
c. Find the value of x that makes this statement true: P(Y ≥ x) = 0.62.

Answer 1

a.
`\( P(-1 \leq Y \leq 1) \)` is the probability that Mr. Shrager dismisses class within 1 minute of its scheduled end time, representing the likelihood of this specific time frame. b.
`\( \mu_Y \)` is the average time after class ends, reflecting the expected value or mean amount of time Mr. Shrager dismisses class.

a. To find
`\( P(-1 \leq Y \leq 1) \)`, you need to integrate the probability density function (pdf) over the interval
`\([-1, 1]\)`:

`\[ P(-1 \leq Y \leq 1) = \int_(-1)^(1) f(y) \, dy \]`

This gives the probability that Mr. Shrager dismisses class within 1 minute of its scheduled end time. The interpretation is the likelihood that the class ends within this specific time frame.

b. The mean
`(\( \mu_Y \))` represents the average amount of time after class ends. It is calculated as:

`\[ \mu_Y = \int_(-\infty)^(\infty) y \cdot f(y) \, dy \]`

This involves multiplying each value of y by its probability density and integrating over the entire range. The interpretation is the expected value or average amount of time Mr. Shrager dismisses class after its scheduled end time.

The complete question is:
Class is over! Mr. Shrager does not always let his statistics class out on time. He seems to end class according to his own "internal clock." The density curve models the distribution of the amount of time after class ends (in minutes) when Mr. Shrager dismisses the class. (A negative value indicates he ended class early.)

A horizontal line is drawn above the axis. A vertical line is drawn from 0 that intersects the horizontal line. The region below the horizontal line between negative 1 and 4 is shaded.

a. Find and interpret P (-1 ≤Y≤ 1).

b. What is μY? Explain your answer.