Question

1) Use worksheet “baseball stats” to perform a multiple regression analysis on the dataset found in BB2011 tab, using Wins as the dependent variable, and League, ERA, Runs Scored, Hits Allowed, Walks Allowed, Saves, and Errors as candidates for the independent variables. Perform the analysis at the 5% significance level.

Team	League	Wins	E.R.A.	Runs Scored	Hits Allowed	Walks Allowed	Saves	Errors
Baltimore	0	69	4.89	708	1568	535	32	110
Boston	0	90	4.20	875	1366	540	36	92
Chicago White Sox	0	79	4.10	654	1463	439	42	79
Cleveland	0	80	4.23	704	1482	463	38	110
Detroit	0	95	4.04	787	1406	492	52	103
Kansas City	0	71	4.44	730	1487	557	37	95
Los Angeles Angels	0	86	3.57	667	1388	476	39	93
Minnesota	0	63	4.58	619	1564	480	32	119
New York Yankees	0	97	3.73	867	1423	507	47	102
Oakland	0	74	3.71	645	1380	519	39	124
Seattle	0	67	3.90	556	1369	436	39	108
Tampa Bay	0	91	3.58	707	1263	504	32	73
Texas	0	96	3.79	855	1327	461	38	114
Toronto	0	81	4.32	743	1433	540	33	110
Arizona	1	94	3.80	731	1414	442	58	90
Atlanta	1	89	3.48	641	1332	521	52	83
Chicago Cubs	1	71	4.33	654	1439	580	40	134
Cincinnati	1	79	4.16	735	1414	539	39	91
Colorado	1	73	4.43	735	1471	522	41	98
Houston	1	56	4.51	615	1477	560	25	116
Los Angeles Dodgers	1	82	3.54	644	1287	507	40	85
Miami	1	72	3.95	625	1403	500	40	93
Milwaukee	1	96	3.63	721	1348	440	47	111
New York Mets	1	77	4.19	718	1482	514	43	116
Philadelphia	1	102	3.02	713	1320	404	47	74
Pittsburgh	1	72	4.04	610	1513	535	43	112
St. Louis	1	90	3.74	762	1461	448	47	116
San Diego	1	71	3.42	593	1324	521	44	94
San Francisco	1	86	3.20	570	1260	559	52	104
Washington	1	80	3.58	624	1403	477	49	104

a) Create a full write up, where you write your statistical analysis step by step. In your write up, make sure you address the following points.

·        Methodology and steps that you took to get to your final regression equation.

·        Final regression equation output.

·        What is the final regression equation?

·        Interpret all the coefficients in the equation.

·        Speak to whether the signs on the coefficients make sense.

·        Interpret R squared.

·        Include a full residual analysis.

·        What is the residual of the Tampa Bay observation?

b) Now, use tab BB2012 to make predictions of wins in 2012, using the model you created with the 2011 stats.  

·        How many games are the Giants (SFG) expected to win in 2012?

·        Which team is predicted by the model to have the worst record in 2012?

·        Which team is predicted by the model to have the best record in 2012?

Team	WINS	E.R.A.	Runs Scored	Hits Allowed	Walks Allowed	Saves	Errors
ARI	81	3.93	734	1432	417	39	91
ATL	94	3.42	700	1310	464	47	89
BAL	93	3.9	712	1433	481	55	109
BOS	69	4.7	734	1449	529	35	100
CHC	61	4.51	613	1399	573	28	105
CHW	85	4.02	748	1365	503	37	70
CIN	97	3.34	669	1356	427	56	89
CLE	68	4.78	667	1503	543	43	97
COL	64	5.22	758	1637	566	36	122
DET	88	3.75	726	1409	438	40	97
HOU	55	4.56	583	1493	540	31	118
KCR	72	4.3	676	1504	542	44	113
LAA	89	4.02	767	1339	483	38	97
LAD	86	3.34	637	1277	539	40	97
MIA	69	4.09	609	1448	495	38	104
MIL	83	4.22	776	1458	525	44	99
MIN	66	4.77	701	1536	465	35	107
NYM	74	4.09	650	1368	488	36	102
NYY	95	3.84	804	1401	431	51	76
OAK	94	3.48	713	1360	462	47	110
PHI	81	3.83	684	1387	409	42	100
PIT	79	3.86	651	1357	490	45	112
SDP	76	4.01	651	1356	539	43	123
SEA	75	3.76	619	1359	449	43	71
SFG	94	3.68	718	1361	489	53	112
STL	88	3.71	765	1420	436	42	107
TBR	90	3.19	697	1233	469	50	113
TEX	93	3.99	808	1378	446	43	86
TOR	73	4.64	716	1439	574	29	102
WSN	98	3.33	731	1296	497	51	96

Answer 1

solution:

Regression line is given as

where

y= dependent/ response variable

= overall mean effect

=effect due to variable x₁

=effect due to variable x₂

and so on

= random error/ residual effect and

here, the dependent variable or y= Wins

we can minimise the error by using method of least squares, ie,

hence solving further, we get Sum of squares to construct ANOVA and calculate R² values which are as follows:

ANOVA
	df	SS	MS	F	Significance F
Regression	7	3465.793	495.1132	34.78095	1.80939E-10
Residual	22	313.174	14.23518
Total	29	3778.967

The R values are:

Regression Statistics89
Multiple R	0.957668
R Square	0.917127
Adjusted R Square	0.890758
Standard Error	3.772954
Observations	30

The regression line is given as:

Wins = 93.70706 - 1.45486 League +-9.65899 E.R.A + 0.084965 Runs Scored -0.02043 Hits Scored -0.03499 Walks Allowed + 0.425586 Saves - 0.04132 Errors +

Here, R²=0.917127 and adjusted R²=0.89075

that indicates that 91%of response variable is explained by the variables of the model.

Now, let us interpret the significance of coefficients

the hypothesis is H₀₁= there is no significance difference between effects of League and similar null hypothesis for other variables respectively. The alrenative hypothesis H₁₁= there is significance between effects of League and similar alternative hypothesis for other variables respectively.

Hence, the decision criteria is to reject H₀ if the p-value < 0.05

The following table tells about the coefficients and p-values and confidence intervals of the model :

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	93.70706	20.93209	4.476717	0.000188279	50.29655	137.1176	50.29655	137.1176
League	-1.45486	1.720024	-0.84584	0.406749342	-5.02197	2.11225	-5.02197	2.11225
E.R.A.	-9.65899	4.368845	-2.21088	0.037743681	-18.7194	-0.59856	-18.7194	-0.59856
Runs Scored	0.084965	0.0097	8.759181	1.26918E-08	0.064848	0.105082	0.064848	0.105082
Hits Allowed	-0.02043	0.019296	-1.05872	0.301213942	-0.06045	0.019588	-0.06045	0.019588
Walks Allowed	-0.03499	0.020262	-1.72707	0.098172005	-0.07702	0.007027	-0.07702	0.007027
Saves	0.425586	0.143276	2.970398	0.007062248	0.12845	0.722723	0.12845	0.722723
Errors	-0.04132	0.054939	-0.75211	0.459958838	-0.15526	0.072616	-0.15526	0.072616

Here, the variables, League, Hit Allowed, Walks Allowed and Errors are significant and ERA, Runs Scored and Saved are insignificant.

The signs of the coefficients do not tell anything about the significance or insignificance of the variable.

The residual for the variables as per the ANOVA table is 14.23518