Question

All work needs to be shown

Consider all observations as one sample of X (1st column) and Y (2nd column) values. Answer the following questions: (20 points)

78	4.4
74	3.9
68	4
76	4
80	3.5
84	4.1
50	2.3
93	4.7
55	1.7
76	4.9
58	1.7
74	4.6
75	3.4
80	4.3
56	1.7
80	3.9
69	3.7
57	3.1
90	4
42	1.8
91	4.1
51	1.8

a) Calculate the correlation coefficient r

b) Fit the regression model (predicting Y from X) and report the estimated intercept and slope.

c) Test whether the slope equals 0. Report your hypothesis, test statistic, p-value.  

All work needs to be shown

Answer 1

The given data is:

X	Y
78	4.4
74	3.9
68	4
76	4
80	3.5
84	4.1
50	2.3
93	4.7
55	1.7
76	4.9
58	1.7
74	4.6
75	3.4
80	4.3
56	1.7
80	3.9
69	3.7
57	3.1
90	4
42	1.8
91	4.1
51	1.8

• 
  
  
  The correlation coefficient is calculated as:
  
  
  
• Let the regression equation be:
  
  
  The estimates of slope is:
  
  
  
  
  
  Hence, the estimated regression equation is:
  
  
  
• We are to test
  
  
  The test statistic is given by:
  
  where,
  
  
  
  We observe the fitted and residual as:

  X	Y
  78	4.4	3.899528	0.250473
  74	3.9	3.643185	0.065954
  68	4	3.258672	0.549567
  76	4	3.771356	0.052278
  80	3.5	4.027699	0.278466
  84	4.1	4.284041	0.033871
  50	2.3	2.105131	0.037974
  93	4.7	4.860811	0.02586
  55	1.7	2.425559	0.526436
  76	4.9	3.771356	1.273836
  58	1.7	2.617816	0.842386
  74	4.6	3.643185	0.915494
  75	3.4	3.707271	0.094415
  80	4.3	4.027699	0.074148
  56	1.7	2.489645	0.623539
  80	3.9	4.027699	0.016307
  69	3.7	3.322757	0.142312
  57	3.1	2.55373	0.29841
  90	4	4.668555	0.446965
  42	1.8	1.592447	0.043078
  91	4.1	4.73264	0.400234
  51	1.8	2.169217	0.136321

  
  Now,
  
  
  We reject H0 at 5% level of significance iff
  
  Now,
  
  
  
  p-value
  
  
  
  
  We reject H0 at 5% level of significance and conclude that   is significant.
  
  

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