Question

equal-areas point.

1. an internet reaction time test asks subjects to click their mouse button as soon as a light flashes on the screen. the light is programmed to go on at a randomly selected time from 2 to 5 seconds after the subject clicks \start\. the density curve models the amount of time the subject has to wait for the light to flash.

(image of a density curve: a rectangle from x=2 to x=5 on the time axis)
time (sec) until
the light flashes
a) what height must the density curve have? (hint: the area under a density curve is always 1)
b) about what proportion of the time will the light turn on between 2.5 and 4 seconds after the subject clicks \start\?

Explanation:

Step1: Calculate the height of the density curve

The density curve is a rectangle. The base of the rectangle is \(b = 5 - 2=3\). Since the area \(A\) under a density curve is \(1\), and for a rectangle \(A=\text{base}\times\text{height}\). Let \(h\) be the height. Then \(A = b\times h\), so \(1=3\times h\). Solving for \(h\), we get \(h=\frac{1}{3}\).

Step2: Calculate the proportion for part (b)

The base for the interval \([2.5,4]\) is \(b = 4 - 2.5 = 1.5\). The height \(h=\frac{1}{3}\). Using the formula for the area of a rectangle \(A=\text{base}\times\text{height}\), we substitute \(b = 1.5\) and \(h=\frac{1}{3}\). Then \(A=1.5\times\frac{1}{3}=\frac{1.5}{3}=0.5\)

Answer:

a) The height of the density curve is \(\frac{1}{3}\)
b) The proportion of the time is \(0.5\)