Question

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 after the subject clicks \start.\ the density curve models the amount of time the subject has to wait for the light to flash
about what percent of the time will the light flash more than 3.75 seconds after the subject clicks \start\?
31.25%
75%
25%
68.75%
option 1
option 2
option 3
option 4

Explanation:

Step1: Determine the total range of the uniform distribution

The uniform distribution ranges from \(a = 0.5\) to \(b = 4.5\) (by observing the graph). The probability density function \(f(x)=\frac{1}{b - a}=\frac{1}{4.5-0.5}=\frac{1}{4}\)

Step2: Calculate the probability using the formula for the area of a rectangle in a uniform distribution

The formula for the probability \(P(X>x)\) in a uniform distribution \(U(a,b)\) is \(P(X > x)=\frac{b - x}{b - a}\) when \(a

$$P(X>3.75)=\frac{4.5 - 3.75}{4.5-0.5}=\frac{0.75}{4}=0.1875$$

Wait, no. Wait, another approach: The area of the rectangle for \(x>3.75\). The base of the rectangle for the region \(x > 3.75\) is \(b-x=4.5 - 3.75 = 0.75\) and the height \(h=\frac{1}{4}\). Area (probability) \(A=\text{base}\times\text{height}\). But actually, since it's a uniform distribution over \([0.5,4.5]\), the length of the interval is \(L=4.5 - 0.5=4\). The length of the sub - interval \((3.75,4.5)\) is \(4.5-3.75 = 0.75\). Probability \(P=\frac{4.5 - 3.75}{4.5 - 0.5}=\frac{0.75}{4}=0.1875\) is wrong. Wait, no! Wait, looking at the graph again (assuming it's a uniform distribution from \(x = 0.5\) to \(x = 4.5\)). The formula for the area of a rectangle (probability) in a uniform distribution \(P(X>x)=\frac{b - x}{b - a}\) (for \(x\in(a,b)\)).
Another way: The total length of the interval where the distribution is non - zero is \(4.5-0.5 = 4\). The length of the interval we are interested in (where \(X>3.75\)) is \(4.5 - 3.75=0.75\). But wait, no! Wait, if we assume the distribution is from \(x = 1\) to \(x = 5\) (maybe mis - reading the x - axis). If it's from \(x = 1\) to \(x = 5\), then \(b - a=4\), \(b - x=5 - 3.75 = 1.25\)

$$P(X>3.75)=\frac{5 - 3.75}{5 - 1}=\frac{1.25}{4}=0.3125$$

Answer:

31.25%