CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 7. Sampling Distributions and Estimations

TABLE7.21:

Hours of Tutoring,x Probability,p

1 0. 2

2 0. 3

3 0. 2

4 0. 3

Solution:

Xwill represent the number of hours per week taking piano lessons andYwill represent the number of hours tutoring
per week.

E(X) =μx=∑xipi Var(X) =σ^2 x=∑(xi−μx)^2 pi

μx= 0 ( 0. 3 )+ 1 ( 0. 3 )+ 2 ( 0. 4 ) σ^2 x= ( 0 − 1. 1 )^2 ( 0. 3 )+( 1 − 1. 1 )^2 ( 0. 3 )+( 2 − 1. 1 )^2 ( 0. 4 )

μx= 1. 1 σ^2 x= 0. 69

σx= 0. 831

E(Y) =μy=∑yipi

μy= 1 ( 0. 2 )+ 2 ( 0. 3 )+ 3 ( 0. 2 )+ 4 ( 0. 3 )

μy= 2. 6

Var(Y) =σ^2 y=∑(yi−μy)^2 pi

σ^2 y= ( 1 − 2. 6 )^2 ( 0. 2 )+( 2 − 2. 6 )^2 ( 0. 3 )+( 3 − 2. 6 )^2 ( 0. 2 )+( 4 − 2. 6 )^2 ( 0. 3 )

σ^2 y= 1. 24

σy= 1. 11

The expected number of hours spent on piano lessons is 1.1 with a standard deviation of 0.831 hours. Likewise, the
expected number of hours I spend tutoring is 2.6 with a standard deviation of 1.11 hours.

I spend $20 for each hour of piano lessons so my mean weekly cost for piano lessons is

μ 20 x= 20 ·μx= ( 20 )( 1. 1 ) =$22.00 Linear Transformation Rule

I earn $25 for each hour of tutoring, so my mean weekly earnings from tutoring are

μ 25 x= 25 ·μy= ( 25 )( 2. 6 ) =$65.00 Linear Transformation Rule

My expected weekly savings are

μ 25 y−μ 20 x=$65. 00 −$22. 00 =$43.00 Subtraction Rule

The standard deviation of the cost of my piano lessons is

σ 20 x= ( 20 )( 0. 831 ) =$16.62 Linear Transformation Rule

The standard deviation of my earnings from tutoring is

σ 25 y= ( 25 )( 1. 11 ) =$27.75 Linear Transformation Rule

The variance of my weekly savings is