Central Limit Theorem: Statistics Review with Exercises, Study notes of Mathematics

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Central Limit Theorem: Review

(modified R. Bloom)

∗

Roberta Bloom

Based on Central Limit Theorem: Review†^ by

Susan Dean

Barbara Illowsky, Ph.D.

This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License ‡

Abstract Practice problems in elementary statistics. One problem in the original module by S. Dean and Dr. B. Illowsky has been revised. The next three questions refer to the following information: Richard's Furniture Company delivers furniture from 10 A.M. to 2 P.M. continuously and uniformly. We are interested in how long (in hours) past the 10 A.M. start time that individuals wait for their delivery.

Exercise 1: REVIEW QUESTION 1 (Solution on p. 4.) X ∼

A. U (0, 4) B. U ( 10 , 2) C. Exp (2) D. N (2, 1)

Exercise 2: REVIEW QUESTION 2 (Solution on p. 4.) The average wait time is:

A. 1 hour B. 2 hour C. 2.5 hour D. 4 hour

Exercise 3: REVIEW QUESTION 3 (Solution on p. 4.) Suppose that it is now past noon on a delivery day. The probability that a person must wait at least 1 12 more hours is: ∗Version 1.2: Dec 18, 2008 6:46 pm US/Central †http://cnx.org/content/m16955/1.8/ ‡http://creativecommons.org/licenses/by/2.0/

A. 14

B. 12

C. 34

D. 38

Exercise 4: REVIEW QUESTION 4 (Solution on p. 4.) Given: X~Exp

( 1

3

)

.

a. Find P (X > 1) b. Calculate the minimum value for the upper quartile. c. Find P

(

X = 13

)

Exercise 5: REVIEW QUESTION 5 (Solution on p. 4.)

• 40% of full-time students took 4 years to graduate
• 30% of full-time students took 5 years to graduate
• 20% of full-time students took 6 years to graduate
• 10% of full-time students took 7 years to graduate

The expected time for full-time students to graduate is:

A. 4 years B. 4.5 years C. 5 years D. 5.5 years

Exercise 6: REVIEW QUESTION 6 (Solution on p. 4.) Which of the following distributions is described by the following example? Many people can run a short distance of under 2 miles, but as the distance increases, fewer people can run that far.

A. Binomial B. Uniform C. Exponential D. Normal

Exercise 7: REVIEW QUESTION 7 (Solution on p. 4.) The length of time to brush one's teeth is generally thought to be exponentially distributed with a mean of 34 minutes. Find the probability that a randomly selected person brushes his/her teeth less than 34 minutes.

A. 0. B. (^34) C. 0. D. 0.

Exercise 8: REVIEW QUESTION 8 (Solution on p. 4.) Which distribution accurately describes the following situation? A 2008 report on technology use states that approximately 20 percent of U.S. households have never sent an e-mail. (source: http://www.webguild.org/2008/05/20-percent-of-americans-have- never-used-email.php) Suppose that we select a random sample of fourteen U.S. households. X =the number of households in the sample of 14 households that have never sent an email

A. B ( 14 , 0 .20)

B. P (2.8)

C. N (2. 8 , 2 .24)

D. Exp

( 1

)

Solutions to Exercises in this Module

Solution to Exercise 1 (p. 1) REVIEW QUESTION 1 Solution : A Solution to Exercise 2 (p. 1) REVIEW QUESTION 1 Solution : B Solution to Exercise 3 (p. 1) REVIEW QUESTION 3 Solution : A Solution to Exercise 4 (p. 2)

REVIEW QUESTION 4 Solution

a. 0. b. 4. c. 0

Solution to Exercise 5 (p. 2) REVIEW QUESTION 5 Solution : C Solution to Exercise 6 (p. 2) REVIEW QUESTION 6 Solution : C Solution to Exercise 7 (p. 2) REVIEW QUESTION 7 Solution : D Solution to Exercise 8 (p. 2) REVIEW QUESTION 8 Solution : A