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Essentials of Statistics for the B...

8th Edition
Frederick J Gravetter + 1 other
ISBN: 9781133956570

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BuyFindarrow_forward

Essentials of Statistics for the B...

8th Edition
Frederick J Gravetter + 1 other
ISBN: 9781133956570
Textbook Problem

There appears to be some evidence suggesting that earlier retirement may lead to memory decline (Rohwedder & Willis. 2010). The researchers gave a memory lest to men and women aged 60 to 64 years in several countries that have different retirement ages. For each country, the researchers recorded the average memory score and the percentage of individuals in the 60 to 64 age range who were retired. Note that a higher percentage retired indicates a younger retirement age for that country. The following data are similar to the results from the study. Use the data to find the regression equation for predicting memory scores from the percentage of people aged 60 to 64 who are retired.

Country % Retired (X) Memory Score (Y)
Sweden 39 9.3
U.S.A. 48 10.9
England 59 10.7
Germany 70 9.1
Spain 74 6.4
Netherlands 78 9.1
Italy 81 7.2
France 87 7.9
Belgium 88 8.5
Austria 91 9.0

To determine
The regression equation of the given data.

Answer

The regression equation of the data is Y=7.004X+133.205.

Explanation

Given info: the given data are shown below,

Country % Retired (X) Memory Score (Y)
Sweden 39 9.3
U.S.A. 48 10.9
England 59 10.7
Germany 70 9.1
Spain 74 6.4
Netherlands 78 9.1
Italy 81 7.2
France 87 7.9
Belgium 88 8.5
Austria 91 9

Calculation:

Generally, the linear regression equation is defined as

Y=bX+a

Formula to calculate b is,

b=SPSSx

Formula to calculate the a is,

a=MYbMX

Mean of the variable X is,

MX=39+48+59+70+74+78+81+87+88+9110=71510=71.5

Thus the mean of the variable X  is 71.5.

Mean of the variable Y is,

MY=9.3+10.9+10.7+9.1+6.4+9.1+7.2+7.9+8.5+910=88.110=8.81

Thus the mean of the variable y is 8.81.

For calculating SSx and SP the table is shown below,

Sr.no X Y XMX YMY (XMX)2 (YMY)2 (XMX)(YMY)
1 39 9.3 -32.5 0.49 1056.25 0.2401 -15.925
2 48 10.9 -23.5 2.09 552.25 4.3681 -49.115
3 59 10.7 -12.5 1.89 156.25 3.5721 -23.625
4 70 9.1 -1.5 0.29 2.25 0.0841 -0.435
5 74 6.4 2.5 -2.41 6.25 5.8081 -6.025
6 78 9.1 6.5 0.29 42.25 0.0841 1.885
7 81 7.2 9.5 -1.61 90.25 2.5921 -15.295
8 87 7.9 15.5 -0.91 240.25 0.8281 -14.105
9 88 8.5 16.5 -0.31 272.25 0.0961 -5.115
10 91 9 19.5 0.19 380.25 0.0361 3.705
Total     0 0 2798.5 17.709 -124.05

Where, MY is the mean of the variable Y and MX is the mean of variable X. Substitute -124.05 for SP and 17.709 for SSX in the equation for calculating b to get the slop coefficient.

b=124.0517.709=7.004

Substitute 7.004 for b, 71.5 for MX and 8.81 for MY In the equation for calculating a to get the coefficient.

a=71.58.81×7.004=71.5+61.705=133.205

Substitute 7.004 for b and 133.205 for a in regression equation to get the regression equation.

Y=7.004X+133.205

The, the regression equation is Y=7.004X+133.205_.

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