Chapter 14.4, Problem 34E

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Retirement benefits The table gives the approximate benefits for PepsiCo executives who earned an average of $250,000 per year during the last 5 years of service, based on the number of years of service, from 15 years to 45 years.(a) Use linear regression to find the linear equation that is the best fit for the data.(b) Use the equation to find the expected annual retirement benefits after 38 years of service.(c) Write a sentence that interprets the slope of the linear regression line. Years of Service Annual Retirement 25$109,280 30 121,130 35 132,990 40 145,490 45 160,790 Source: TRICON Salaried Employees Retirement Plan

(a)

To determine

To calculate: The equation of the line that is the best fit for the data and the data for the year of service and the annual retirement for PepsiCo executives from 15 years to 45 years is tabulated below.

 Years of service Annual retirement 25 $109,280 30 121,130 35 132,990 40 145,490 45 160,790 Explanation Given Information: The data for the year of service and the annual retirement for PepsiCo executives from 15 years to 45 years is tabulated below.  Years of service Annual retirement 25$109,280 30 121,130 35 132,990 40 145,490 45 160,790

Formula used:

The equation of the best fit line by linear regression that for the given data points (x1,y1),(x2,y2),......,(xn,yn) is y^=a+bx,

Where b=âˆ‘i=1nxiâˆ‘i=1nyiâˆ’nâˆ‘i=1nxiyi(âˆ‘i=1nxi)2âˆ’nâˆ‘i=1nxi2, and a=âˆ‘i=1nyiâˆ’bâˆ‘i=1nxin.

Calculation:

Consider the table,

 Years of service Annual retirement 25 $109,280 30 121,130 35 132,990 40 145,490 45 160,790 Let xi denote the years of service for 1<i<5. That is, x1=25,x2=30,x3=35,x4=40, x5=45. Let yi denote the values of annual retirement for 1<i<5. That is, y1=109,280,y2=121,130,y3=132,990,y4=145,490, y5=160,790. Since, the number of data points is 5. Thus, n=5. The value of b for a best fit line from a linear regression is, b=âˆ‘i=1nxiâˆ‘i=1nyiâˆ’nâˆ‘i=1nxiyi(âˆ‘i=1nxi)2âˆ’nâˆ‘i=1nxi2, Substitute 5 for n (b) To determine To calculate: The expected annual retirement benefits after 38 year of service and the data for the year of service and the annual retirement for PepsiCo executives from 15 years to 45 years is tabulated below.  Years of service Annual retirement 25$109,280 30 121,130 35 132,990 40 145,490 45 160,790

(c)

To determine

To calculate: The slope of the linear regression line and interpret it and the data for the year of service and the annual retirement for PepsiCo executives from 15 years to 45 years is tabulated below.

 Years of service Annual retirement 25 \$109,280 30 121,130 35 132,990 40 145,490 45 160,790

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