Question

Let x be a random variable that represents the length of time it takes a student to complete a take home exam in a geology class. It was found that x has a normal distribution with mean µ = 5.2 hours and standard deviation σ = 1.8 hours.

1. Convert the x interval x ≥ 9.7 to a standard z interval.

2. Convert the z interval -1.5 ≤ z ≤ 1 to a raw score x interval

Answer 1

Let, X be a random variable that represents the length of time it takes a student to complete a take home exam in a geology class.

X follows normal distribution with mean 5.2 hours and standard deviation 1.8 hours.

So, Z=(X-5.2)/1.8~normal(0,1), ie. standard normal distribution.

So, X=1.8*Z+5.2.

(a) To convert the X interval X>=9.7 to a standard Z interval.

X>9.7

ie. X-5.2>9.7-5.2

ie. X-5.2>4.5

ie. (X-5.2)/1.8>4.5/1.8

ie. Z>4.5/1.8

Where Z is the standard normal variate.

ie. Z>2.5.

So, the corresponding standardised Z interval is Z>2.5.

(b) Convert the Z interval -1.5 ≤ Z≤ 1 to a raw score X interval.

-1.5 ≤ Z≤ 1

ie. -1.5*1.8<=Z*1.8<=1*1.8

ie. -2.7<=Z*1.8<=1.8

ie. -2.7+5.2<=X<=1.8+5.2

ie. 2.5<=X<=7.0

So, the corresponding raw score X interval is -2.5 ≤ X≤ 7.0.