Question

For a normal distribution with a mean of µ = 150 and an SD of σ = 15:

3. Find these probabilities: a. p (X > 150) b. p(X < 120) c. p(X < 170) d. p(130 < X < 175)

A researcher wants to test her hypothesis that drinking caffeine while learning a new skill will aid in developing that skill. In order to test her hypotheses, she recruits a sample of 25 beginner piano students from a nearby college (n = 25). Each piano player is given a new song to learn and asked to drink a large coffee each time they sit down to practice it over a 6 week span of time. Afterward, when the six weeks are up, they are asked to perform the song for an instructor who rates their progress using a standardized scoring system. For the general population of beginner piano players, scores using this system are normally distributed with a mean (or µ) of 70 and a standard deviation (or σ) of 15. For this particular group of 25 caffeine-drinking beginner piano players, the mean score was 75.

Answer 1

This is a normal distribution question with


a) P(x > 150.0)=?
The z-score at x = 150.0 is,

z = 0.0
This implies that
P(x > 150.0) = P(z > 0.0) = 1 - 0.5


b) P(x < 120.0)=?
The z-score at x = 120.0 is,

z = -2.0
This implies that


c P(x < 170.0)=?
The z-score at x = 170.0 is,

z = 1.3333
This implies that


d) P(130.0 < x < 175.0)=?

This implies that
P(130.0 < x < 175.0) = P(-1.3333 < z < 1.6667) = P(Z < 1.6667) - P(Z < -1.3333)
P(130.0 < x < 175.0) = 0.9522129635397043 - 0.09121668684984685

PS: you have to refer z score table to find the final probabilities.