Ivo Dinov
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STAT 13 (1a, 1b, 1c)

Introduction to Statistical Methods for the Life and Health Science

Department of Statistics

## Instructor: Ivo Dinov

Homework 4
Due Date: Wednesday, Nov. 10, 2004

Please, submit your homework before lecture on the due date. See the HW submission rules. On the front page include the following header.

• (HW_4_1)
• Suppose X~ Normal(μ = 23, σ2 = 4). Use SOCR (http://socr.stat.ucla.edu/) Distribution-Modeler to calculate the following probabilities. Include a snapshot of each area of interest from your work using the SOCR pages.
1. P(X  ≤ 21.5);
2. P(X < 21.5);
3. P(X > 22.4);
4. P(24 ≤ X ≤ 27);
5. P(20 ≤ X ≤ 27  ∩ 23 ≤ X ≤ 29)
6. P(20 ≤ X ≤ 23  ∪ 24 ≤ X ≤ 27)

• 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 = 9.1 liters and standard deviation σ = 2.78 liters. Use the SOCR (http://socr.stat.ucla.edu/) resource to calculate:
1. What is the least amount of soft serve ice cream that is needed so that the driver can satisfy demand on 90% of afternoons?
2. What is the interquartile range for the ice cream sales.

• Use either SOCR or STATA to solve the following problems where X~Normal(μ = 4.1, σ2 = 0.82 ):

1. What is the probability that X is greater than 5.1?
2. What is the probability that X is between 4.3 and 5.6?
3. What value of x gives P(X ≤x) = 0.45?

• X has a mean of -2 and a standard deviation of 4.5 and W has a mean of 4.5 and a standard deviation of 3.1. Let X and W be independent random variables and let Y = 3X - 3W.
1. What are the mean and standard deviation of Y?
2. What can we say about the shape of the distribution of Y?

• (HW_4_2) Suppose that X~ Normal(μ=2.1, σ2=9) compute the Z-scores for the following numbers and state how many SD's each of these numbers is away from μ:
• -2
• 9
• 6
• -0.4
• What is the probability P(-1.2 ≤ X ≤ 0)? Is it different from P(-1.2 < X < 0)? Why?

• (HW_4_3) Suppose the height of males is normally distributed with a mean of 71" (inches) and a standard deviation of 2.9". Assume also that the height of women is normally distributed with mean of 65.7" and standard deviation of 2.6". Calculate and represent graphically the following probabilities.
• What is the chance that a randomly selected male will be 72.1" to 73.8" 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 137"?
• Under the above independence assumption, what is the chance that the female is taller than her male partner?

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