Question

For the two variables of interest: Create a scatter plot with Percent Time Asleep as the independent variable x and Longevity as the dependent variable y. The plot must include an informative title, along with correct labels for both axes. Include a plot of the least-squares equation (see #5 below). Calculate the correlation coefficient and the coefficient of determination. Identify any data points on the scatter diagram that appear to be influential. Use Cook's Distance > (4⁄√n) as the criterion for an influential data point. If there are no influential points, say so. Conduct a formal hypothesis test at α=0.05 to determine if there is evidence of linear correlation between the two variables. Present the results in four parts similar to those used for the hypothesis test in Section 4 (omit the distribution type): Your null and alternate hypotheses in the proper format. The P-value and its logical relationship to α (≤ or >). Your decision regarding the null hypothesis: reject or fail to reject. A statement regarding the sufficiency of the evidence for correlation. Construct the least-squares equation (must be in algebraic format for full credit). Determine if the equation you constructed in #5 above is a valid model. Justify your decision with a detailed analysis that includes an assessment of the coefficient of determination (#2 above), a discussion of the effect of the hypothesis test results (#4 above) on model validity, and an assessment of the residuals, to include a residual histogram, a Q-Q plot, and a plot of the residuals against Percent Time Asleep (with explanatory paragraphs for each graphic).

Pct Time Asleep	Longevity
22	35
9	37
49	49
1	46
23	63
83	39
23	46
15	56
9	63
81	65
12	56
15	65
37	70
24	63
26	65
17	70
14	77
14	81
6	86
25	70
18	70
26	77
24	77
29	81
27	77
18	40
6	37
19	44
7	47
16	47
13	47
35	68
2	47
35	54
6	61
15	71
14	75
18	89
50	58
25	59
10	62
33	79
43	96
35	58
17	62
27	70
22	72
16	75
20	96
37	75
23	46
4	42
20	65
42	46
9	58
32	42
66	48
28	58
10	50
4	80
12	63
17	65
12	70
23	70
40	72
18	97
10	46
38	56
7	70
23	70
36	72
9	76
21	90
62	76
36	92
23	21
62	40
28	44
18	54
10	36
28	40
22	56
29	60
15	48
73	53
10	60
5	60
13	65
27	68
20	60
21	81
12	81
49	48
17	48
22	56
71	68
17	75
10	81
24	48
18	68
34	16
6	19
4	19
22	32
28	33
31	33
16	30
27	42
8	42
32	33
20	26
35	30
12	40
14	54
17	34
29	34
31	47
6	47
30	42
27	47
40	54
19	54
8	56
8	60
15	44

Answer 1

1.

Steps(excel): Enter data > selet data > insert > scatterplot > add trendline and display euqation and R-sq > OK

Scatterplot of Longevity vs Pct Time 120 100 y = 0.0046x + 57.332 R?= 2E-05 80 Longevity 40 20 0 0 10 20 30 60 70 80 90 40 50 Pct Time Asleep

2. Regression:

Steps(excel): Enter data > data > Data anlysis > regression > enter y range and x range > ok

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.004178	correlation
R Square	1.75E-05	determination
Adjusted R Square	-0.00811
Standard Error	17.63499
Observations	125
ANOVA
	df	SS	MS	F	Significance F
Regression	1	0.667661	0.667661	0.002147	0.963119	> 0.05 (not significant)
Residual	123	38252.13	310.9929
Total	124	38252.8
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	57.33157	2.822116	20.3151	4.09E-41	51.74536	62.91778
X Variable 1	0.004621	0.099735	0.046334	0.963119	-0.1928	0.202039

Model is not significant.

only intercept is significant as it has p-value < 0.05.

3.

RESIDUAL OUTPUT
Observation	Predicted Y	Residuals	Standard Residuals
1	57.43323	-22.4332	-1.27725
2	57.37316	-20.3732	-1.15996
3	57.55801	-8.55801	-0.48725
4	57.33619	-11.3362	-0.64543
5	57.43786	5.562144	0.316683
6	57.71512	-18.7151	-1.06555
7	57.43786	-11.4379	-0.65122
8	57.40089	-1.40089	-0.07976
9	57.37316	5.62684	0.320367
10	57.70588	7.294119	0.415294
11	57.38702	-1.38702	-0.07897
12	57.40089	7.599113	0.432659
13	57.50255	12.49745	0.711548
14	57.44248	5.557523	0.31642
15	57.45172	7.548281	0.429765
16	57.41013	12.58987	0.71681
17	57.39627	19.60373	1.116148
18	57.39627	23.60373	1.34389
19	57.3593	28.6407	1.630672
20	57.4471	12.5529	0.714706
21	57.41475	12.58525	0.716547
22	57.45172	19.54828	1.112991
23	57.44248	19.55752	1.113517
24	57.46558	23.53442	1.339943
25	57.45634	19.54366	1.112728
26	57.41475	-17.4148	-0.99152
27	57.3593	-20.3593	-1.15917
28	57.41937	-13.4194	-0.76404
29	57.36392	-10.3639	-0.59007
30	57.40551	-10.4055	-0.59244
31	57.39164	-10.3916	-0.59165
32	57.49331	10.50669	0.598204
33	57.34081	-10.3408	-0.58876
34	57.49331	-3.49331	-0.19889
35	57.3593	3.640703	0.207285
36	57.40089	13.59911	0.774272
37	57.39627	17.60373	1.002277
38	57.41475	31.58525	1.798321
39	57.56263	0.437374	0.024902
40	57.4471	1.552902	0.088415
41	57.37778	4.622219	0.263168
42	57.48407	21.51593	1.22502
43	57.53028	38.46972	2.190292
44	57.49331	0.506691	0.028849
45	57.41013	4.589871	0.261327
46	57.45634	12.54366	0.714179
47	57.43323	14.56677	0.829366
48	57.40551	17.59449	1.001751
49	57.42399	38.57601	2.196344
50	57.50255	17.49745	0.996226
51	57.43786	-11.4379	-0.65122
52	57.35005	-15.3501	-0.87396
53	57.42399	7.576008	0.431344
54	57.52566	-11.5257	-0.65622
55	57.37316	0.62684	0.035689
56	57.47945	-15.4794	-0.88133
57	57.63656	-9.63656	-0.54866
58	57.46096	0.539039	0.03069
59	57.37778	-7.37778	-0.42006
60	57.35005	22.64995	1.289586
61	57.38702	5.612977	0.319578
62	57.41013	7.589871	0.432133
63	57.38702	12.61298	0.718126
64	57.43786	12.56214	0.715232
65	57.51641	14.48359	0.82463
66	57.41475	39.58525	2.253805
67	57.37778	-11.3778	-0.6478
68	57.50717	-1.50717	-0.08581
69	57.36392	12.63608	0.719441
70	57.43786	12.56214	0.715232
71	57.49793	14.50207	0.825682
72	57.37316	18.62684	1.060528
73	57.42861	32.57139	1.854468
74	57.61808	18.38192	1.046584
75	57.49793	34.50207	1.964392
76	57.43786	-36.4379	-2.07461
77	57.61808	-17.6181	-1.00309
78	57.46096	-13.461	-0.76641
79	57.41475	-3.41475	-0.19442
80	57.37778	-21.3778	-1.21715
81	57.46096	-17.461	-0.99415
82	57.43323	-1.43323	-0.0816
83	57.46558	2.534417	0.144298
84	57.40089	-9.40089	-0.53524
85	57.66891	-4.66891	-0.26583
86	57.37778	2.622219	0.149297
87	57.35468	2.645325	0.150613
88	57.39164	7.608355	0.433185
89	57.45634	10.54366	0.600308
90	57.42399	2.576008	0.146666
91	57.42861	23.57139	1.342048
92	57.38702	23.61298	1.344416
93	57.55801	-9.55801	-0.54419
94	57.41013	-9.41013	-0.53577
95	57.43323	-1.43323	-0.0816
96	57.65967	10.34033	0.588732
97	57.41013	17.58987	1.001488
98	57.37778	23.62222	1.344942
99	57.44248	-9.44248	-0.53761
100	57.41475	10.58525	0.602676
101	57.48869	-41.4887	-2.36218
102	57.3593	-38.3593	-2.18401
103	57.35005	-38.3501	-2.18348
104	57.43323	-25.4332	-1.44805
105	57.46096	-24.461	-1.3927
106	57.47482	-24.4748	-1.39349
107	57.40551	-27.4055	-1.56035
108	57.45634	-15.4563	-0.88001
109	57.36854	-15.3685	-0.87502
110	57.47945	-24.4794	-1.39375
111	57.42399	-31.424	-1.78914
112	57.49331	-27.4933	-1.56534
113	57.38702	-17.387	-0.98994
114	57.39627	-3.39627	-0.19337
115	57.41013	-23.4101	-1.33287
116	57.46558	-23.4656	-1.33602
117	57.47482	-10.4748	-0.59639
118	57.3593	-10.3593	-0.58981
119	57.4702	-15.4702	-0.8808
120	57.45634	-10.4563	-0.59534
121	57.51641	-3.51641	-0.20021
122	57.41937	-3.41937	-0.19468
123	57.36854	-1.36854	-0.07792
124	57.36854	2.631461	0.149824
125	57.40089	-13.4009	-0.76299

4.

Plots:

Q-Q plot:

Normal Probability Plot 150 100 Y 50 0 0 20 100 120 40 60 80 Sample Percentile

Y is approximately Normal as the line is straight.

5. Residual vs Predicted:

Residuals vs Predicted Y 50 40 30 20 10 . . 0 57.3 -10 57.35 524 57.45 37.5 57.55 57.6 57.65 57.7 57.75 -20 -30 -40 . -50

Pattern is random so there is no heteroscedasticity.

6.

Cooks distance:

D, = Σ(- Y5%)? (p + 1)σ?

Now, the model is not significant at all there doesnt seem to be an observation/(s) that once removed will enhance the model. So, we do not reset the model or calculate Cook's distance as there doesnt seem to be any neither model seem to be significant for the changes.

Please rate my answer and comment for doubt.