Comparison Of Mean Tablet Weight For Three Different Batches In An Essay. (70 Characters)

Descriptive Statistics for Batch 1

Question:

Using SPSS, compare the three batches in terms of their arithmetic mean tablet weight.Explain the choice of the statistical test(S) employed and provide detailed results and discussion.

We have to compare the three batches in terms of their arithmetic mean tablet weight. For this comparison purpose, we have to see some descriptive statistics for these three batches. We have to use the one way ANOVA test for comparison of means of three batches. We have to use the SPSS for statistical analysis purpose. Let us see all this comparison in detail.

First we have to see the descriptive statistics for the first batch. We know that descriptive statistics consist of the mean, standard deviation, variance, minimum, kurtosis, etc. All descriptive statistics for the first batch is given in the following table:

Descriptive Statistics
	N	Minimum	Sum	Mean	Std. Deviation	Variance
Batch1	20	.35	7.37	.3686	.01546	.000
Valid N (listwise)	20

Some other descriptive statistics for the first batch are given in the following table:

Descriptive Statistics
	N	Range	Maximum	Mean	Skewness		Kurtosis
	Statistic	Statistic	Statistic	Std. Error	Statistic	Std. Error	Statistic	Std. Error
Batch1	20	.05	.39	.00346	.103	.512	-1.418	.992
Valid N (listwise)	20

The box plot for the weights of first batch is given below:

Now, let us see the descriptive statistics for the weights of second batch. All descriptive statistics for the weights of second batch are given in the following two tables:

Descriptive Statistics
	N	Minimum	Sum	Mean	Std. Deviation	Variance
Batch2	20	.35	7.40	.3699	.01636	.000
Valid N (listwise)	20

Descriptive Statistics
	N	Range	Maximum	Mean	Skewness		Kurtosis
	Statistic	Statistic	Statistic	Std. Error	Statistic	Std. Error	Statistic	Std. Error
Batch2	20	.05	.39	.00366	-.020	.512	-1.657	.992
Valid N (listwise)	20

The box plot for the weights of the second batch is given as below:

Now, we have to see the some descriptive statistics for the weights of the third batch. The descriptive statistics for the weights of third batch are given in the following table:

Descriptive Statistics
	N	Minimum	Sum	Mean	Std. Deviation	Variance
Batch3	20	.38	7.95	.3974	.00885	.000
Valid N (listwise)	20

Descriptive Statistics
	N	Range	Maximum	Mean	Skewness		Kurtosis
	Statistic	Statistic	Statistic	Std. Error	Statistic	Std. Error	Statistic	Std. Error
Batch3	20	.03	.41	.00198	-.440	.512	-1.107	.992
Valid N (listwise)	20

The box plot for the weights of the third batch is given as below:

Now, we have to compare these three batches or average weights of these three batches. For comparison of means of weights of these three batches, we have to use the one way ANOVA test. We take significance level for this test as alpha = 0.05.

The null and alternative hypothesis for this test is given as below:

Null hypothesis: H₀: The means of weights for all three batches are same.

Alternative hypothesis: H_a: The means of weights for all three batches are not same.

In statistical words, these hypotheses are written as below:

H₀: µ₁ = µ₂ = µ₃ V/s H_a: µ₁ ≠ µ₂ ≠ µ₃

Where, µ₁ is the mean for weights for the first batch, µ₂ is the mean for weights for the second batch and µ₃ is the mean for weights of third batch.

The ANOVA table by using SPSS is given as below:

ANOVA
Weight
	Sum of Squares	df	Mean Square	F	Sig.
Between Groups	.011	2	.005	27.191	.000
Within Groups	.011	57	.000
Total	.022	59

For this ANOVA table, we get the p-value as 0.000 and we have level of significance or alpha value = 0.05. We know the decision rule is given as below:

We reject the null hypothesis if the p-value is less than the alpha value or level of significance and we do not reject the null hypothesis if the p-value is greater than the alpha value or level of significance.

Here we have alpha value = 0.05 and p-value = 0.05

That is, here p-value < alpha value

So, we reject the null hypothesis that the means of weights for all three batches are same.

Descriptive Statistics for Batch 2

In the next topic we have to compare the means and standard deviations for the tablet tensile strength and tablet porosity. Also we have to find the some intervals for means. Let us see the descriptive statistics for tensile strength and porosity in detail. The means and standard deviations are given in the following table:

Descriptive Statistics
	N	Mean	Std. Deviation
TSB1	10	10.5460	.02066
TSB2	10	10.0500	.00000
TSB3	10	10.4200	.12293
TPB1	10	3.5290	.02234
TPB2	10	3.6650	.15219
TPB3	10	3.6800	.31903
Valid N (listwise)	10

TSB1 = Tensile strength for batch 1

TSB2 = Tensile strength for batch 2

TSB3 = Tensile Strength for batch 3

TPB1 = Tablet porosity for batch 1

TPB2 = Tablet porosity for batch 2

TPB3 = Tablet porosity for batch 3

The overall mean and standard deviation for the strength and porosity is given as below:

Descriptive Statistics
	Mean	Std. Deviation	N
Strength	10.3387	.22508	30
Porosity	3.6247	.20905	30

One standard deviation limits from the mean for the strengths and porosity of these three batches are given as below:

Batch	Tensile strength		Porosity
	Lower	Upper	Lower	Upper
Batch 1	10.52534	10.56666	3.50666	3.55134
Batch 2	10.05	10.05	3.51281	3.81719
Batch 3	10.29707	10.54293	3.36097	3.99903

Now, in the next topic we have to see the relationship between the strength and porosity. We have to check whether there is any linear relationship or association between the strength and porosity exists or not. For this purpose we have to find the correlation coefficient between the two variables strength and porosity.

The SPSS output for the correlation coefficient is given as below:

Correlations
		Strength	Porosity
Strength	Pearson Correlation	1	-.092
	Sig. (2-tailed)		.627
	N	30	30
Porosity	Pearson Correlation	-.092	1
	Sig. (2-tailed)	.627
	N	30	30

The correlation coefficient between the two variables strength and porosity is found as 0.627, this is a positive correlation coefficient. This indicates that there is positive considerable linear relationship or association exists between the two variables strength and porosity.

Let us see regression analysis for above two variables. The SPSS output is given below:

Descriptive Statistics
	Mean	Std. Deviation	N
Strength	10.3387	.22508	30
Porosity	3.6247	.20905	30

Below is the correlation coefficient for these two variables.

Correlations
		Strength	Porosity
Pearson Correlation	Strength	1.000	-.092
	Porosity	-.092	1.000
Sig. (1-tailed)	Strength	.	.314
	Porosity	.314	.
N	Strength	30	30
	Porosity	30	30

The description for the variables is given in the following table:

Variables Entered/Removeda
Model	Variables Entered	Variables Removed	Method
1	Porosityb	.	Enter
a. Dependent Variable: Strength
b. All requested variables entered.

The model summary for the regression analysis is given below:

Model Summaryb
Model	R	R Square	Adjusted R Square	Std. Error of the Estimate	Durbin-Watson
1	.092a	.009	-.027	.22808	.556
a. Predictors: (Constant), Porosity
b. Dependent Variable: Strength

For the above model summary, we get the coefficient of determination as 0.009, this means that only 0.9% of the variation in the dependent variable is explained by the independent variable.

The ANOVA table for the regression analysis is given below:

ANOVAa
Model		Sum of Squares	df	Mean Square	F	Sig.
1	Regression	.013	1	.013	.241	.627b
	Residual	1.457	28	.052
	Total	1.469	29
a. Dependent Variable: Strength
b. Predictors: (Constant), Porosity

For above ANOVA, we get the p-value as 0.627 which is greater than the level of significance or alpha value = 0.05, so we do not reject the null hypothesis that the given regression model is significant.

The coefficients for the regression equation are given below:

Coefficientsa
Model		Unstandardized Coefficients		Standardized Coefficients	t	Sig.	95.0% Confidence Interval for B		Collinearity Statistics
		B	Std. Error	Beta			Lower Bound	Upper Bound	Tolerance	VIF
1	(Constant)	10.699	.736		14.546	.000	9.192	12.206
	Porosity	-.099	.203	-.092	-.491	.627	-.514	.316	1.000	1.000
a. Dependent Variable: Strength

Collinearity Diagnosticsa
Model	Dimension	Eigenvalue	Condition Index	Variance Proportions
				(Constant)	Porosity
1	1	1.998	1.000	.00	.00
	2	.002	35.299	1.00	1.00
a. Dependent Variable: Strength

Residuals Statisticsa
	Minimum	Maximum	Mean	Std. Deviation	N
Predicted Value	10.2914	10.3908	10.3387	.02079	30
Residual	-.30007	.28871	.00000	.22412	30
Std. Predicted Value	-2.274	2.510	.000	1.000	30
Std. Residual	-1.316	1.266	.000	.983	30
a. Dependent Variable: Strength

References:

•    Abramowitz, M., and Stegun, I.A., "Handbook of mathematical functions", Dover publications, New York, 1964

•    Crow, E.L., Davis, F.A., and Maxfield, M.W. "Statistics Manual", Dover publications, Inc New York, 1960

•    Fraser, D.A.S., "Nonparametric methods in statistics", John Wiley&Sons, New York, Chapman&Hall, London, 1957.

•    Kanji, G.K., "100 statistical tests", Sage publications London, Newbury Park, New Dehli.

•    Papoulis, Athanasios "Probability and Statistics" Prentence-Hall International Editions.

•    Siegel, S. "Non-parametric statistics for the behavioral sciences", McGraw-Hill book company, Inc. New York, Toronto, London, 1956

•    Van den Brink, W.P., and Koele, P. "Statistiek, Deel 3: Toepassingen", Boom Meppel Amsterdam

•    Abraham, B., & Ledolter, J. (1983). Statistical methods for forecasting. New York: Wiley.

•    Adorno, T. W., Frenkel-Brunswik, E., Levinson, D. J., & Sanford, R. N. (1950). The authoritarian personality. New York: Harper.

•    Agrawal, R., Imielinski, T., & Swami, A. (1993). Mining association rules between sets of items in large databases. Proceedings of the 1993 ACM SIGMOD Conference, Washington, DC.

•    Agrawal, R. & Srikant, R. (1994). Fast algorithms for mining association rules. Proceedings of the 20th VLDB Conference. Santiago, Chile.

•    Bain, L. J. (1978). Statistical analysis of reliability and life-testing models. New York: Decker.

•    Bain, L. J. and Engelhardt, M. (1989) Introduction to Probability and Mathematical Statistics. Kent, MA: PWS.

•    Baird, J. C. (1970). Psychophysical analysis of visual space. New York: Pergamon Press.