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: H_{0}: 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:
H_{0}: µ_{1} = µ_{2} = µ_{3} V/s H_{a}: µ_{1} ≠ µ_{2} ≠ µ_{3}
Where, µ_{1} is the mean for weights for the first batch, µ_{2} is the mean for weights for the second batch and µ_{3} 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 pvalue 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 pvalue is less than the alpha value or level of significance and we do not reject the null hypothesis if the pvalue is greater than the alpha value or level of significance.
Here we have alpha value = 0.05 and pvalue = 0.05
That is, here pvalue < alpha value
So, we reject the null hypothesis that the means of weights for all three batches are same.
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. (2tailed) 
.627 

N 
30 
30 

Porosity  Pearson Correlation 
.092 
1 
Sig. (2tailed) 
.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. (1tailed)  Strength 
. 
.314 
Porosity 
.314 
. 

N  Strength 
30 
30 
Porosity 
30 
30 
The description for the variables is given in the following table:
Variables Entered/Removed^{a} 

Model 
Variables Entered 
Variables Removed 
Method 
1  Porosity^{b} 
. 
Enter 
a. Dependent Variable: Strength  
b. All requested variables entered. 
The model summary for the regression analysis is given below:
Model Summary^{b} 

Model 
R 
R Square 
Adjusted R Square 
Std. Error of the Estimate 
DurbinWatson 
1 
.092^{a} 
.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:
ANOVA^{a} 

Model 
Sum of Squares 
df 
Mean Square 
F 
Sig. 

1  Regression 
.013 
1 
.013 
.241 
.627^{b} 
Residual 
1.457 
28 
.052 

Total 
1.469 
29 

a. Dependent Variable: Strength  
b. Predictors: (Constant), Porosity 
For above ANOVA, we get the pvalue 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:
Coefficients^{a} 

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 Diagnostics^{a} 

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 Statistics^{a} 

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 
Chart
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