Statistics - normal distribution question?
PS I had to replace some mathematical symbols with words as I could not reproduce them here.
The Hamilton (HAM-D) Scale is used to assess patients for clinical depression. In clinically depressed patients, the scale is normally distributed with a mean of 19 points, and a standard deviation of 4 points.
a. Borderline patients have HAM-D scores between 7 and 10. What proportion of depressed patients fall in this range?
X=HAM-D score and we know X~N(19,16)
We want P(7<X<10).
Transforming we get:
P(7-19/ √16 < x- mu/standard deviation < 10-19/√16) = P(-3<Z<-2.25)
b. The cutoff for severe depression is the 90th percentile; what raw score does this correspond to on the HAM-D scale?
90th percentile in the Z distribution corresponds to 1.282.
z=x-mu/standard deviation=1.282. Solving for x we get:
c. Consider a random sample of 16 patients. What is the distribution of the sample mean for this sample?
x̄ ~ N(19, 16/16)
e. What is the probability that the sample mean of the 16 patients is between 17 and 20?
P(17-19/1 < x- mu/variance /n < 20-19/1) = P(-2<Z< 1)