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Continuous Random Variables: Review

Module by: Barbara Illowsky, Ph.D., Susan Dean. E-mail the authors

Summary: This module provides a number of homework/review problems related to Continuous Random Variables. Note: This module is currently under revision, and its content is subject to change. This module is being prepared as part of a statistics textbook that will be available for the Fall 2008 semester.

Note: You are viewing an old version of this document. The latest version is available here.

Exercise 1Exercise 7 refer to the following study: A recent study of mothers of junior high school children in Santa Clara County reported that 76% of the mothers are employed in paid positions. Of those mothers who are employed, 64% work full-time (over 35 hours per week), and 36% work part-time. However, out of all of the mothers in the population, 49% work full-time. The population under study is made up of mothers of junior high school children in Santa Clara County.

Let E = E = size 12{E={}} {} employed, Let F = F = size 12{F={}} {} full-time employment

Exercise 1

  • a. Find the percent of all mothers in the population that NOT employed.
  • b. Find the percent of mothers in the population that are employed part-time.

Solution

  • a. 24%
  • b. 27%

Exercise 2

The type of employment is considered to be what type of data?

Solution

Qualitative

Exercise 3

In symbols, what does the 36% represent?

Solution

P ( PT E ) P ( PT E ) size 12{P \( ital "PT" \lline E \) } {}

Exercise 4

Find the probability that a randomly selected person from the population will be employed OR work full-time.

Solution

0.7336

Exercise 5

Based upon the above information, are being employed AND working part-time:

  • a. mutually exclusive events? Why or why not?
  • b. independent events? Why or why not?

Solution

  • a. No,
  • b. No,

Exercise 6 - Exercise 7 refer to the following: We randomly pick 10 mothers from the above population. We are interested in the number of the mothers that are employed. Let X = X = size 12{X={}} {} number of mothers that are employed.

Exercise 6

State the distribution for X X size 12{X} {} .

Solution

B ( 10 , 0 . 76 ) B ( 10 , 0 . 76 ) size 12{B \( "10",0 "." "76" \) } {}

Exercise 7

Find the probability that at least 6 are employed.

Solution

0.9330

Exercise 8

We expect the Statistics Discussion Board to have, on average, 14 questions posted to it per week. We are interested in the number of questions posted to it per day.

  • a. Define X X size 12{X} {} .
  • b. What are the values that the random variable may take on?
  • c. State the distribution for X X size 12{X} {} .
  • d. Find the probability that from 10 to 14 (inclusive) questions are posted to the Listserv on a randomly picked day.

Solution

  • a. X=X= size 12{X={}} {} the number of questions posted to the Statistics Listserv per day
  • b. x = 0,1,2, . . . x = 0,1,2, . . . size 12{x=0,1,2, "." "." "." } {}
  • c. X ~ P ( 2 ) X ~ P ( 2 ) size 12{X "~" P \( 2 \) } {}
  • d. 0

Exercise 9

A person invests $1000 in stock of a company that hopes to go public in 1 year.

  • The probability that the person will lose all his money after 1 year (i.e. his stock will be worthless) is 35%.
  • The probability that the person’s stock will still have a value of $1000 after 1 year (i.e. no profit and no loss) is 60%.
  • The probability that the person’s stock will increase in value by $10,000 after 1 year (i.e. will be worth $11,000) is 5%.

Find the expected PROFIT after 1 year.

Solution

$150

Exercise 10

Rachel’s piano cost $3000. The average cost for a piano is $4000 with a standard deviation of $2500. Becca’s guitar cost $550. The average cost for a guitar is $500 with a standard deviation of $200. Matt’s drums cost $600. The average cost for drums is $700 with a standard deviation of $100. Whose cost was lowest when compared to his or her own instrument? Justify your answer.

Solution

Matt

Exercise 11

For the following data, which of the measures of central tendency would be the LEAST useful: mean, median, mode? Explain why. Which would be the MOST useful? Explain why.

4,6,6, 12 , 18 , 18 , 18 , 200 4,6,6, 12 , 18 , 18 , 18 , 200 size 12{4,6,6,"12","18","18","18","200"} {}

Solution

Mean

Exercise 12

Horizontal boxplot with first whisker extending from 1 to 2, box from 2 to 5,  line at 4, and second whisker extending from 5 to 7.

For each statement below, explain why each is either true or false.

  • a. 25% of the data are at most 5.
  • b. There is the same amount of data from 4 – 5 as there is from 5 – 7.
  • c. There are no data values of 3.
  • d. 50% of the data are 4.

Solution

  • a. False
  • b. True
  • c. False
  • d. False

Exercise 13Exercise 14 refer to the following: 64 faculty members were asked the number of cars they owned (including spouse and children’s cars). The results are given in the following graph: Histogram consisting of 5 bars with number of cars, from 0-7 in increments of 1, on the x-axis, and frequency, in increments of 0.1 from 0.15-0.45, on the y-axis. No bars exist for 4, 5, or 7. Bar 0 has a frequency of 0.075, 1 has 0.15, 2 has 0.45, 3 has 0.25, and 6 has 0.075.

Exercise 13

Find the approximate number of responses that were “3.”

Solution

16

Exercise 14

Find the first, second and third quartiles. Use them to construct a box plot of the data.

Solution

2,2,3 2,2,3 size 12{2,2,3} {}

Exercise 15Exercise 16 refer to the following study done of the Girls soccer team “Snow Leopards”:

Table 1
Hair Style   Hair Color  
  blond brown black
ponytail 3 2 5
plain 2 2 1
Suppose that one girl from the Snow Leopards is randomly selected.

Exercise 15

Find the probability that the girl has black hair GIVEN that she wears a ponytail.

Solution

5 10 = 0 . 5 5 10 = 0 . 5 size 12{ { {5} over {"10"} } =0 "." 5} {}

Exercise 16

Find the probability that the girl wears her hair plain OR has brown hair.

Solution

7 15 7 15 size 12{ { {7} over {"15"} } } {}

Exercise 17

Find the probability that the girl has blond hair AND that she wears her hair plain.

Solution

2 15 2 15 size 12{ { {2} over {"15"} } } {}

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