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Continuous Random Variables: Review

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.

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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.

• a. 24%
• b. 27%

Exercise 2

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

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.

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.

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.

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"} {}

Mean

Exercise 12

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:

Exercise 13

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

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”:

 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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