Connect Hosted by ALEKS Access Card or Elementary Statistics
Connect Hosted by ALEKS Access Card or Elementary Statistics
3rd Edition
ISBN: 9781260373752
Author: William Navidi Prof., Barry Monk Professor
Publisher: McGraw-Hill Education
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Chapter 4, Problem 3CS
To determine

To predict the unemployment in a year when inflation is 3%

Expert Solution & Answer
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Answer to Problem 3CS

The unemployment rate is 5.97% when the inflation rate is 3%

Explanation of Solution

Given:

The inflation rate and the unemployment rate, both in percent, for the years 1988-2015 is as shown below.

    YearInflationUnemployment
    19884.45.5
    19894.65.3
    19906.15.6
    19913.16.8
    19922.97.5
    19932.76.9
    19942.76.1
    19952.55.6
    19963.35.4
    19971.74.9
    19981.64.5
    19992.74.2
    20003.44.0
    20011.64.7
    20022.45.8
    20031.96.0
    20043.35.5
    20053.45.1
    20062.54.6
    20074.14.6
    20080.15.8
    20092.79.3
    20101.59.6
    20113.08.9
    20121.78.1
    20131.57.4
    20140.86.2
    20150.75.3

Concept used:

Given ordered pairs (x,y) with sample means x¯andy¯ , sample standard deviations sxandsy and the correlation coefficient γ , the equation of the least squares regression line for predicting y from x is,

  y^=b0+b1x , where b1=γsysxistheslpoeandb0=y¯b1x¯istheyintercept.

First we find the sample means x¯andy¯ .

  x¯=xn=72.928=2.603571y¯=yn=169.228=6.042857

Now to find standard deviations, we construct the table as shown below:

    xy(xx¯)2(yy¯)2
    4.45.53.2271570.294693
    4.65.33.9857280.551836
    6.15.612.2250150.196122
    3.16.80.2464410.573265
    2.97.50.0878702.123265
    2.76.90.0092980.734694
    2.76.10.0092980.003265
    2.55.60.0107260.196122
    3.35.40.4850130.413265
    1.74.90.8164401.306122
    1.64.51.0071542.380407
    2.74.20.0092983.396121
    3.44.00.6342994.173264
    1.64.71.0071541.803264
    2.45.80.0414410.058979
    1.96.00.4950120.001836
    3.35.50.4850130.294693
    3.45.10.6342990.888979
    2.54.60.0107262.081836
    4.14.62.2392992.081836
    0.15.86.2678670.058979
    2.79.30.00929810.608980
    1.59.61.21786812.653266
    3.08.90.1571558.163266
    1.78.10.8164404.231837
    1.57.41.2178681.841837
    0.86.23.2528680.024693
    0.75.33.6235820.551836
    ( x x ¯ )2= 44.229617 ( y y ¯ )2= 61.688558

Therefore, standard deviations are,

  sx= ( x x ¯ ) 2 n1= 44.229627 281= 44.229617 27=1.6381339=1.2798sy= ( y y ¯ ) 2 n1= 61.688558 281= 61.688558 27=2.2847614=1.5115

Now to find correlation coefficient, we construct the table as shown below:

    xyxx¯sxyy¯sy(xx¯sx)(yy¯sy)
    4.45.51.4036794 0.3591511 0.5041330
    4.65.31.5599538 0.4914700 0.7666704
    6.15.62.7320120 0.2929917 0.8004568
    3.16.80.38789570.50092160.1943053
    2.97.50.23162130.96403770.2232916
    2.76.90.07534690.56708100.0427277
    2.76.10.07534690.03780540.0028485
    2.55.6 0.0809274 0.2929917 0.0237110
    3.35.40.5441701 0.4253106 0.2314413
    1.74.9 0.7060251 0.7561078 0.5338310
    1.64.5 0.7841623 1.0207456 0.8004302
    2.74.20.0753469 1.2192239 0.0918647
    3.44.00.6223073 1.3515428 0.8410749
    1.64.7 0.7841623 0.8884267 0.6966707
    2.45.8 0.1590646 0.1606728 0.0255573
    1.96.0 0.5497507 0.0283539 0.0155875
    3.35.50.5441701 0.3591511 0.1954392
    3.45.10.6223073 0.6237889 0.3881883
    2.54.6 0.0809274 0.9545861 0.0772521
    4.14.61.1692678 0.9545861 1.1161667
    0.15.8 1.9562205 0.1606728 0.3143114
    2.79.30.07534692.15490770.1623656
    1.59.6 0.8622995 2.3533860 2.0293235
    3.08.90.30975851.89026990.5855271
    1.78.1 0.7060251 1.3609943 0.9608961
    1.57.4 0.8622995 0.8978782 0.7742399
    0.86.2 1.4092600 0.1039649 0.1465135
    0.75.3 1.4873972 0.4914700 0.7310111
    ( x x ¯ s x )( y y ¯ s y ) =

       4.4169802

Therefore,

  γ=( x x ¯ s x )( y y ¯ s y )n1=4.4169802281=4.416980227=0.1635910.16

  b1=γsysx=(0.16)(1.51151.2798)=0.1932

  &b0=y¯b1x¯=6.042857(0.1932)2.603571=6.042857+0.503033=6.54589

Hence least squares regression line is,

  y^=6.54580.1932x

Now substituting x=3 in the least squares regression line, we get

  y^=6.54580.1932(3)=6.54580.5796=5.96625.97

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Chapter 4 Solutions

Connect Hosted by ALEKS Access Card or Elementary Statistics

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