Question

0. The table below shows data for 13 students in a statistics class. Each member of the class ran a 40-yard sprin and then did a long iump (with a running start). Sprint Time (s) 5.41 5.05 9.49 8.09 7.01 7.17 6.83 6.73 8.01 5.68 5.78 6.31 6.04 Long Jump (in) 171 184 48 151 90 65 94 78 71 130 173 143 141 a) Create and label a scatterplot of the data (Sprint time vs Long Jump) b) Describe and interpret the scatterplot above. c) Calculate the LSRL equation and the regression coefficient r and R2. What can you say about them? d) Plot the Residuals vs Long jump. e) Calculate the LSRL and regression coefficient of the Residuals.

Answer 1

a)

b)

Scatterplot shows a negative direction

AS the values of sprint time,X increases, long jump distance,Y gets decrease

c)

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
5.41	171	1.7648101	2768.379	-69.8975
5.05	184	2.8509024	4305.379	-110.789
9.49	48	7.5709639	4953.994	-193.666
8.09	151	1.8266562	1063.763	44.08095
7.01	90	0.0737331	805.6864	-7.70751
7.17	65	0.1862254	2849.917	-23.0375
6.83	94	0.0083793	594.6095	-2.23213
6.73	78	7.16E-05	1630.917	0.341716
8.01	71	1.6168101	2245.302	-60.2514
5.68	130	1.1203408	134.9172	-12.2944
5.78	173	0.9186485	2982.84	-52.3467
6.31	143	0.1835793	605.9172	-10.5467
6.04	141	0.4878485	511.4556	-15.796

.

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	87.6	1539	18.60897	25453.08	-514.142
mean	6.738461538	118.384615	SSxx	SSyy	SSxy

sample size ,   n =   13          
here, x̅ =   6.738461538       ȳ =   118.3846154  
                  
SSxx =    Σ(x-x̅)² =    18.60896923          
SSxy=   Σ(x-x̅)(y-ȳ) =   -514.1423077          
                  
slope ,    ß1 = SSxy/SSxx =   -27.62873651          
                  
intercept,   ß0 = y̅-ß1* x̄ =   304.5597937          
                  
so, regression line is   Ŷ =   304.5598   +   -27.6287   *x
                  
SSE=   (Sx*Sy - S²xy)/Sx =    11247.97          
                  
std error ,Se =    √(SSE/(n-2)) =    31.9772          
                  
correlation coefficient ,    r = Sxy/√(Sx.Sy) =   -0.7471          
                  
R² =    (Sxy)²/(Sx.Sy) =    0.5581          
                  
there is negative correlation between X and Y

and 55.81% variation in variable Y is explained by variable X

d)

Y	Ŷ	Residual
171	155.0883	15.91167
184	165.0347	18.96533
48	42.36308	5.636916
151	81.04332	69.95668
90	110.8824	-20.8824
65	106.4618	-41.4618
94	115.8555	-21.8555
78	118.6184	-40.6184
71	83.25361	-12.2536
130	147.6286	-17.6286
173	144.8657	28.1343
143	130.2225	12.77753
141	137.6822	3.317775

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