One and Two (or more) Sample Hypothesis Testing Paper. Using data from one of the data sets available through the “Data Sets” link on your   page, develop one business research question from which you will formulate a research hypothesis to test one population parameter and another to test two (or more) population parameters.  Formulate both a numerical and verbal hypothesis statement regarding each of your research issue.

Perform hypotheses tests using the five step model. Describe and interpret the results of the test, both in statistical terms and in conversational English. Include appropriate descriptive statistics.

Solution:

Research question: To find whether there is a significant difference between wins and salary of the baseball players.

There are two leagues denoted as

1 if American League and

0 if National League

We have separated the data set as

Data set

American League:

 

 

 

Claim: There is a significant difference between wins and salary- mil of the baseball players in American League.

Hypotheses:

Null Hypothesis:

Numerical Null Hypothesis:

   

Verbal Null Hypothesis:

 There is no significant difference between wins and salary- mil of the baseball players in American League.

Alternative Hypothesis:

Numerical Alternative Hypothesis:

   

Verbal Alternative Hypothesis:

 There is a significant difference between wins and salary- mil of the baseball players in American League.

Level of Significance:

 α = 0.05

Decision rule:

If the p value is greater than the given level of significance we may accept the null hypothesis. Otherwise reject the null hypothesis.

Test Statistic:

                         

 

 

 

 

Using Megastat in Microsoft Excel Add- Ins:

Add- Ins ŕ MegastatŕHypothesis tests ŕ Compare two independent groups

 

Hypothesis Test: Independent Groups (t-test, pooled variance)
	Salary -mil	Wins
	75.479	81.71	mean
	45.930	13.07	std. dev.
	14	14	n
		26	df
		-6.2357	difference (Salary -mil - Wins)
		1,140.1793	pooled variance
		33.7665	pooled std. dev.
		12.7626	standard error of difference
		0	hypothesized difference
		-0.49	t
		.6292	p-value (two-tailed)

 

The test statistic value is -0.49.

The p value for the test statistic is 0.6292.

 

Conclusion:

Since the p value of test statistic is greater than 0.05 level of significance we may accept the null hypothesis H₀ at 5% level of significance. Hence, we conclude that there is no significant difference between wins and salary- mil of the baseball players in American League.

 

 

 

Research question: To find whether there is a significant difference between wins and salary of the baseball players.

 

Data set

National League:

 

Claim: There is a significant difference between wins and salary- mil of the baseball players in National League.

Hypotheses:

Null Hypothesis:

Numerical Null Hypothesis:

   

Verbal Null Hypothesis:

 There is no significant difference between wins and salary- mil of the baseball players in National League.

 

 

Alternative Hypothesis:

Numerical Alternative Hypothesis:

   

Verbal Alternative Hypothesis:

 There is a significant difference between wins and salary- mil of the baseball players in National League.

Level of Significance:

 α = 0.05

Decision rule:

If the p value is greater than the given level of significance we may accept the null hypothesis. Otherwise reject the null hypothesis.

Test Statistic:

                         

 

Using Megastat in Microsoft Excel Add- Ins:

Add- Ins ŕ MegastatŕHypothesis tests ŕ Compare two independent groups

 

Hypothesis Test: Independent Groups (t-test, pooled variance)
	Salary -mil	Wins
	70.949	80.375	mean
	20.669	8.831	std. dev.
	16	16	n
		30	df
		-9.4257	difference (Salary -mil - Wins)
		252.5883	pooled variance
		15.8930	pooled std. dev.
		5.6190	standard error of difference
		0	hypothesized difference
		-1.68	t
		.1038	p-value (two-tailed)

 

The test statistic value is -1.68.

The p value for the test statistic is 0.1038.

 

Conclusion:

Since the p value of test statistic is greater than 0.05 level of significance we may accept the null hypothesis H₀ at 5% level of significance. Hence, we conclude that there is no significant difference between wins and salary- mil of the baseball players in National League.

 

Regression analysis:

The general multiple regression is given by         

where, y is the dependent variable,

          ’s are independent variable,

           is the actual constant,

           is the actual coefficient associated with ith independent variable,

           is the error term which models the unsystematic error of the y    

The above model can be written in matrix form as    

                             

The General Goal of multiple regression is to determine which independent (explanatory) variables should be included in the model.

We want to first test each coefficient,  where i=1,2,...,k, within the model, in order to determine if that individual parameter should be dropped from the model.

Next we test the goodness of fit of the model.

 

 

Hypothesis Tests:

      

   

Procedure:

 First we estimate the model as

    

 where,  is the estimated value of  and .

 

 For Testing Each :

The test statistic is given by 

where,  is the standard error of the estimated coefficient .

Goodness of fit test:           

In order to test the goodness of fit test we generally compute R², which lies between 0 and 1. As R² tends to 1, we can say that the model is suitable for the data i.e. the model can explain the data very well. 

 

 

 

 

 

 

 

Dependent variable:

X7- Wins

Independent variables:

X2- League

X3- Built

X4- Size

X5- Surface

X6- Salary- mil

X8- Attendance

X9- Batting

X10- ERA

X11- HR

X12- Error

X13- SB

Using Megastat in Microsoft Excel Add- Ins:

Add- Ins ŕ MegastatŕCorrelation/ Regression ŕ Regression analysis

Regression Analysis
	R˛	0.857
	Adjusted R˛	0.770	n	30
	R	0.926	k	11
	Std. Error	5.200	Dep. Var.	Wins
ANOVA table
Source	SS	df	MS	F	p-value
Regression	2,917.2794	11	265.2072	9.81	1.64E-05
Residual	486.7206	18	27.0400
Total	3,404.0000	29
Regression output					confidence interval
variables	coefficients	std. error	t (df=18)	p-value	95% lower	95% upper
Intercept	74.6634	133.9145	0.558	.5840	-206.6805	356.0073
League	-1.2494	2.3275	-0.537	.5980	-6.1392	3.6404
Built	-0.0274	0.0558	-0.491	.6291	-0.1447	0.0899
Size	-0.00000401	0.00020556	-0.019	.9847	-0.00043588	0.00042787
Surface	0.5761	4.3135	0.134	.8952	-8.4863	9.6384
Salary -mil	0.0411	0.0667	0.615	.5462	-0.0992	0.1813
Attendance	-0.00000085	0.00000317	-0.267	.7923	-0.00000750	0.00000581
Batting	447.7443	200.5131	2.233	.0385	26.4819	869.0067
ERA	-13.6362	2.4171	-5.642	2.37E-05	-18.7143	-8.5581
HR	0.0930	0.0338	2.755	.0130	0.0221	0.1639
Error	-0.1601	0.1246	-1.285	.2151	-0.4218	0.1017
SB	0.0152	0.0361	0.422	.6777	-0.0605	0.0910

 

The regression equation is

Wins = 74.6634 - 1.2494 League - 0.0274 Built - 0.00000401 Size + 0.5761 Surface + 0.0411 Salary -mil - 0.00000085 Attendance + 447.7443 Batting -13.6362 ERA + 0.0930 HR -0.1601 Error + 0.0152 SB

The R-Sq(adj.) value is high. So the model has good fit. But the p-values for x2, x3, x4, x5, x6, x12 and x13 are greater than 0.05. So these coefficients are insignificant. There is thus a multicollinearity problem. So we drop these variables and regress x7 on x9, x10 and x11.

 

Regression Analysis: x7 versus x9, x10, x11

 

Dependent variable:

X7- Wins

Independent variables:

X9- Batting

X10- ERA

X11- HR

Using Megastat in Microsoft Excel Add- Ins:

Add- Ins ŕ MegastatŕCorrelation/ Regression ŕ Regression analysis

 

 

The regression equation is

Wins = 1.8499 + 492.4490 Batting -15.9575 ERA + 0.1035 HR

Here all the p values of the coefficients are less than 0.05 i.e.  are significant at 5 % level of significance. The R² value is slightly reduced after dropping the variables and it is of not that much effect and hence the model is good.

 

 

Correlation:

Research question: To find whether salary have relationship with Attendance of the baseball players.

There are two leagues denoted as

1 if American League and

0 if National League

We have separated the data set as

Data set

American League:

 

Using Megastat in Microsoft Excel Add- Ins:

Add- Ins ŕ MegastatŕCorrelation/ Regression ŕ Correlation Matrix

Correlation Matrix
		Salary -mil	Attendance
	Salary -mil	1.000
	Attendance	.895	1.000
		14	sample size

The correlation coefficient between salary- mil and attendance is 0.895. there is a strong positive correlation exist between the variables.

 

Null Hypothesis:

H₀: ρ=0

H₀: “no linear relationship” between the variables.

Alternative Hypothesis:

H₁: ρ≠0

H₁:“ linear relationship” between the variables.

Level of significance:

α = 0.05

Critical value:

At 5% level of significance t distribution with v = 14 - 2 degrees of freedom is  2.178813

Test statistic:

Under

     has a t distribution with v = n-2 degrees of freedom.

  r= 0.895 and n = 14

   

 

Conclusion:

Since the test statistic value is greater than the critical value there is no evidence to accept the null hypothesis at 5% level of significance. Hence we conclude that there is a relationship exist between the variables salary- mil and attendance.

 

Data set

National League:

 

Using Megastat in Microsoft Excel Add- Ins:

Add- Ins ŕ MegastatŕCorrelation/ Regression ŕ Correlation Matrix

 

Correlation Matrix
		Salary -mil	Attendance
	Salary -mil	1.000
	Attendance	.693	1.000
		16	sample size

The correlation coefficient between salary- mil and attendance is 0.693. There is a strong positive correlation exist between the variables.

Null Hypothesis:

H₀: ρ=0

H₀: “no linear relationship” between the variables.

Alternative Hypothesis:

H₁: ρ≠0

H₁:“ linear relationship” between the variables.

Level of significance:

α = 0.05

Critical value:

At 5% level of significance t distribution with v = 16 - 2 degrees of freedom is  2.144787

Test statistic:

Under

      has a t distribution with v = n-2 degrees of freedom.

   r= 0.693 and n = 16

    

 

 

Conclusion:

Since the test statistic value is greater than the critical value there is no evidence to accept the null hypothesis at 5% level of significance. Hence we conclude that there is a relationship exist between the variables salary- mil and attendance.

Descriptive Statistics:

Using Megastat in Microsoft Excel Add- Ins:

Add- Ins ŕ MegastatŕDescriptive Statistics

 

The descriptive statistics for the whole team is given in the above table.

 

 

Inference for our research:

 

• From the analysis of comparing two independent groups we obtain the result as there is no significant difference between wins and salary- mil of the baseball players in American League.
• From the analysis of comparing two independent groups we obtain the result as there is no significant difference between wins and salary- mil of the baseball players in National League.
• From the regression analysis we obtained the regression equation predicting the wins is

Wins = 1.8499 + 492.4490 Batting -15.9575 ERA + 0.1035 HR

·         From the correlation analysis we obtained the result as there is a relationship exists between the variables salary- mil and attendance of baseball players in American League.

• From the correlation analysis we obtained the result as there is a relationship exists between the variables salary- mil and attendance of the baseball players in National League.