10. A psychiatrist believes that 80% of all people who visit doctors have proble
ID: 2936244 • Letter: 1
Question
10. A psychiatrist believes that 80% of all people who visit doctors have problems of a psychosomatic nature. She decides to select 25 patients at random to test her theory.
(a) Assuming that the psychiatrist's theory is true, what is the expected value of x, the number of the 25 patients who have psychosomatic problems?
patients
(b) What is the variance of x, assuming that the theory is true?
(c) Find
P(x 15).
(Use tables and assume that the theory is true. Enter your answer to three decimal places.)
P(x 15) =
(d) Based on the probability in part (c), if only 15of the 25 sampled had psychosomatic problems, what conclusions would you make about the psychiatrist's theory? Explain.
Explanation / Answer
NORMAL APPROXIMATION TO BINOMIAL DISTRIBUTION
the PDF of normal distribution is = 1/ * 2 * e ^ -(x-u)^2/ 2^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
a.
mean ( np ) = 25 * 0.8 = 20
b.
variance ( npq )= 25*0.8*0.2 = 4
standard deviation ( npq )= 25*0.8*0.2 = 2
equation of the normal curve is ( Z )= x - u / sd/sqrt(n) ~ N(0,1)
c.
P(X > 15) = (15-20)/2
= -5/2 = -2.5
= P ( Z >-2.5) From Standard Normal Table
= 0.9938
P(X < = 15) = (1 - P(X > 15))
= 1 - 0.0062 = 0.9938
d.
p = 15/25 = 0.60
P(X > 15) = (15-15)/2.4495
= 0/2.4495 = 0
= P ( Z >0) From Standard Normal Table
= 0.5
P(X < = 15) = (1 - P(X > 15))
= 1 - 0.5 = 0.5
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