Ivo Dinov
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STAT 13 (2a, 2b, 2c)

Introduction to Statistical Methods for the Life and Health Science

Department of Statistics

Instructor: Ivo Dinov

Homework 4
Due Date: Friday, Oct. 31, 2003

Please, submit your homework right after lecture on the due date. 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
Use the table to find the following when X~ Normal(μ = 23, σ2 = 4):
(i ) pr(X ≤19);   (ii) pr(X < 19);  (iii) pr(X > 21);
(iv) pr(24 ≤ X ≤ 27)

• 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
The number of liters of soft serve ice cream sold by an ice cream van driver in an afternoon is found to be Normally distributed with mean μ = 8.6 liters and standard deviation σ = 1.28 liters.

(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?

(ii) What is the interquartile range for the ice cream sales.

• Use either SOCR or STATA to solve the following problems where X~Normal(μ = 5.1, σ2 = 0.872 ):

• (i) What is the probability that X is greater than 6?
(ii) What is the probability that X is between 4.3 and 6.6?
(iii) What value of x gives P(X ≤x) = 0.45?
• 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.

•  (i) What are the mean and standard deviation of Y?

(ii) What can we say about the shape of the distribution of 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:
• -6
• 10
• 7
• -0.4
• What is the probability P(-2 ≤ X ≤ 0)? Is it different from P(-2 < X < 0)? Why?

• (HW_4_3) The height of males is normally distributed with a mean of 70" (inches) and a standard deviation of 3". Assume also that the height of women is normally distributed with mean of 65.5" and standard deviation of 2.5". Calculate and represent graphically the following probabilities.
• What is the chance that a randomly selected male will be 73" to 75" tall?
• If heights of male and female partments are independent (a strong and unclear assumption) what is the chance that their combined height is less than 140"?
• Under the above independence assumption, what is the chance that the female is taller than her male partner?

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