STAT 13

(Sec. 1a-1c)

Introduction to Statistical Methods for the Life and Health Sciences

Instructor: Ivo Dinov, Asst. Prof.

Departments of Statistics & Neurology

http://www.stat.ucla.edu/~dinov/courses_students.html

HOMEWORK 4 - Sketch of Solutions

Due Date:

Friday, Feb. 07, 2003, turn in after lecture

See the HW submission rules. On the front page include the following header.

HW_4_1

The following table of Normal probabilities was obtained from Excel.

x	P(X≤x)	x	P(X ≤ x)	x	P(X ≤ x)
15	0.0000	20	0.0668	25	0.8413
16	0.0002	21	0.1587	26	0.9332
17	0.0013	22	0.3085	27	0.9772
18	0.0062	23	0.5000	28	0.9938
19	0.0228	24	0.6915	29	0.9987

The following table of probabilities was obtained from STATA: Normal with mean μ= 8.6 and standard deviation σ= 1.28.

P(X ≤ x)	x
0.1000	6.9596
0.2500	7.7367
0.7500	9.4633
0.9000	10.2404

mean

8.6

standard deviation

1.28

(i) What is the least amount of soft serve ice cream that is needed so that the driver can satisfy demand on 90% of afternoons? Z=(X-8.6)/1.28, Z (90%) = 1.28. Solve for X!

Use either STATA or a graphics calculator to solve the following problems where

X~Normal(μ = 5.1, σ² = 0.87² ):

(iii) What value of x gives P(X ≤x) = 0.45? Z = (X - 5.1)/0.87 = 1.645, Solve for X!

X has a mean of -3 and a standard deviation of 5 and W has a mean of 5 and a standard deviation of 3. Let X and W be independent random variables and let Y = 3X - 3W.

SD(Y) = Sqrt( 225 + 81) = 17.5
(ii) What can we say about the shape of the distribution of Y? If X and W are normally distributed, then so is Y!

(HW_4_2) Suppose that X~ Normal(μ=3, σ²=16) compute the Z-scores for the following numbers and state how many SD's each of these numbers is away from μ:

-5 ; Z = (X-3)/4 = -2 (-2 SD's away from mean, below mean)
11; Z = (11-3)/4 = 2
5; Z = 1
1.4; Z = -3/8
What is the probability P(-3 ≤ X ≤ -1)? Is it different from P(-3 < X < -1)? Why? The two probabilities are the same, since the X ~ Normal Distribution and so X is continuous and including the limits does not change the chance. Z1 = (-3-3)/4 = -3/2, Z2 = (-4)/4 = -1, P(-3<X<-1) = 0.4332 - 0.3413.

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Ivo D. Dinov, Ph.D., Departments of Statistics and Neurology, UCLA School of Medicine