Let x be the age in years of a licensed automobile driver. Let y be the percentage of all fatal accidents (for a given age) due to speeding. For example, the first data pair indicates that 36% of...

Let x be the age in years of a licensed automobile driver. Let y be the percentage of all fatal accidents (for a given age) due to speeding. For example, the first data pair indicates that 36% of all fatal accidents involving 17 year olds are due to speeding.

x=17,27,37,47,57,67,77

y=36,25,20,12,10,7,5

1.)find the sample mean for x (round to nearest whole number) and y (round to 2 decimal places).

2.)find a and b. round your answer to 3 decimal places

3.)write the equation of the least squares line y=a+bx

4.)Predict the percentage of all fatal accidents due to speeding for 25 year olds. (round to 2 decimal places)

 

Expert Answers
embizze eNotes educator| Certified Educator

We are given the following coordinate pairs: (17,36), (27,25), (37,20), (47,12), (57,10), (67,7), and (77,5), where the x-coordinate represents the age in years and the y-coordinate the percent of fatal accidents with speeding as the major cause.

It might be easiest to create a table with the following headings: x, y, xy, x^2, and y^2. Then sum each of the columns:

x:      y:      xy:      x^2:      y^2:
17     36     612     289        1296
27     25     675     729        625
37     20     740    1369       400
47     12     564    2209       144
57     10     570    3249       100
67      7      469    4489         49
77      5      385    5929         25
-----------------------------------
329   115  4015   18263     2639

(1) Find the mean of x and y:

`bar(x)=(sum x)/n=329/7=47 `

`bar(y)=(sum y)/n=115/7 ~~ 16.43 `

(2) Find a and b for the linear regression line of best fit. (Note that you should check that the correlation is significant—with r about -.959 the correlation is significant at least for a 98% confidence.)

`a=((sum y)(sum x^2)-(sum x)(sum xy))/(n(sum x^2)-(sum x)^2) `

`=(115*18263-329*4015)/(7*18263-329^2) ~~ 39.761 `

`b=(n(sum xy)-(sum x)(sum y))/(n(sum x^2)-(sum x)^2) `

`=(7*4015-329*115)/(7*18263-329^2) ~~ -.496 `

(3) The equation of the regression line y'=a+bx is y'=39.760-0.496x

(4) The value of the regression line at x=25 (which is the estimate for the percentage of fatal accidents caused by speeding for 25 year olds) is

y'=39.760-0.496(25)=27.35

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