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# Statistical Analysis of the Relationship Between the Two Variables Report

## Abstract

In the measurement of any relationship between variables, it is essential to use a correlation statistic to determine the strength of relationship between them. A data from three studies carried out on a given sample of population was obtained and analyzed by using both SPSS and Excel.

The mean, standard deviation, range and F-test were obtained from three groups of samples in order to analyze the two variables X and Y. A one sample t-test was obtained at a 95% confidence interval after which the results were interpreted. ANOVA was also conducted in order to provide the value of F-score for interpretation purposes. The results had shown a lower standard deviation for a large sample size implying that a large sample size should be used to test the relationship between variables since it affects reduces variability of data.

## Introduction

This report is aimed at obtaining a statistical analysis of the relationship between the two variables X and Y before determining the effect of changing sample size on variability. The X variable was the independent variable and the Y variable was the dependent variable.

It will also explain the significance of F-score as obtained by ANOVA using SPSS. Three groups have been obtained from different sample sizes to ensure validity and reliability of results obtained in making a conclusion. The experiment involved the selection of various random variables in groups of 30 members for appropriate description and presentation of results.

To determine the measurements for variability, the measures of range, standard deviation, variance and median were used to provide useful information for interpretation.

## Methods

The experiment was been performed on random variables of X and Y obtained from a certain population. First, a sample size of 15 variables was obtained and recorded as group 1, second was group 2 with sample size of 60 and lastly group 3 with sample size of 90. The correlation for 30 variables of each group of sample was recorded for the study.

In total, three groups of samples were obtained in order to ensure reliability of the results. Correlation was then calculated for each group to determine the relationship between X and Y. The correlation coefficient obtained from each group was then recorded for analysis.

ANOVA was also conducted to evaluate whether the average score was equal or different between the means of groups (Adivia, 2010, par.3).

## Results

### Degree of Freedom

For each the studies carried out, the number of data entries n is equal to 30.Therefore it means that the degree of freedom DF=n-1 is DF = 30-1 = 29

Critical Values

### Correlation #1

One-Sample Statistics

 N Mean Std. Deviation Std. Error Mean X 30 52.27 30.007 5.478 Y 30 47.23 30.602 5.587

One-Sample Test

 Test Value = 0 T df Sig. (2-tailed) Mean Difference 95% Confidence Interval of the Difference Lower Upper X 9.540 29 .000 52.27 41.06 63.47 Y 8.454 29 .000 47.23 35.81 58.66

### Correlation #2

One-Sample Statistics

 N Mean Std. Deviation Std. Error Mean X 30 58.13 27.474 5.016 Y 30 49.67 28.779 5.254

One-Sample Test

 Test Value = 0 T df Sig. (2-tailed) Mean Difference 95% Confidence Interval of the Difference Lower Upper X 11.590 29 .000 58.13 47.87 68.39 Y 9.453 29 .000 49.67 38.92 60.41

### Correlation #3 One-Sample Statistics

 N Mean Std. Deviation Std. Error Mean X 30 41.57 27.859 5.086 Y 30 67.17 23.915 4.366

One-Sample Test

 Test Value = 0 T df Sig. (2-tailed) Mean Difference 95% Confidence Interval of the Difference Lower Upper X 8.172 29 .000 41.57 31.16 51.97 Y 15.383 29 .000 67.17 58.24 76.10

Using SPSS, one-tail and critical two-tail t values were obtained using a one sample t test for each study and the results were as follows:

Study Group #1: 9.540 and 8.454

Study Group #2: 11.590 and 9.453

Study Group #3: 8.172 and 15.383

Standard Deviation

In SPSS the standard deviation for each study was obtained using one sample t test and the results were as follows:

σ1: 30.602

σ2: 28.779

σ3: 23.915

### Range of Variation

Using SPS, the range of variation was obtained by using Analyze>Descriptive Statistics>Descriptives>Range

 Descriptive Statistics N Range Minimum Maximum Group 1 30 96 2 98 Group 2 30 92 3 95 Group 3 30 8 11 19

### Correlation Mean

In SPSS the mean correlation for each study was obtained using one sample t test and the results were provided as follows:

Group #1=0.022

Group #2=0.122

Group #3= -0.128

### ANOVA Results Group 1 ANOVA Y

 Sum of Squares Df Mean Square F Sig. Between Groups 24368.367 25 974.735 1.398 .411 Within Groups 2789.000 4 697.250 Total 27157.367 29

From the results obtained in the above ANOVA table, the F-score of 1.398 is greater than the significance value of the F test in the Group 1 ANOVA table which is 0.411. We reject the null hypothesis and conclude that average assessment score differs across the groups of variable X and Y.

### Group 2 ANOVA Y

 Sum of Squares Df Mean Square F Sig. Between Groups 22209.667 23 965.638 3.203 .076 Within Groups 1809.000 6 301.500 Total 24018.667 29

From the results obtained in the Group 2 ANOVA table, the F-score 3.203 is less than significance value of the F test in the table which is 0.76. We reject the null hypothesis and conclude that average assessment score is different across the groups of variable X and Y

### Group 3 ANOVA Y

 Sum of Squares Df Mean Square F Sig. Between Groups 14144.167 26 544.006 .668 .761 Within Groups 2442.000 3 814.000 Total 16586.167 29

From the results obtained in the Group 3 ANOVA table, the F-score 0.668 is less than significance value of the F test in the Group 3 ANOVA table which is 0.761. We accept the null hypothesis and conclude that average assessment is equal across the groups of variable X and Y

## Affects of Changes Sample Size on Variability

When the sample size was small as given by Group #1, the standard deviation was 30.602, while Group #2 and #3 had 28.779 and 23.915 respectively. This shows that the standard deviation decreases with the increase in sample size.

The sample size selected for any population affects the confidence interval of the data. If the sampling size is increased, the required confidence interval will also increase. The reason why the confidence interval increases is because of many variables that reduce the variance from one variable to another (Ramsey, 2009, par. 2).

From the results obtained earlier in the ANOVA table, it is clear that Group #1 which had started with a smaller sample size had shown the average assessment scores was different across the groups within the significance value of 0.411. As the samples size was increased, there was a no much difference in scores between the variables X and Y as shown by Group #2 and #3.

The correlation mean obtained for the three groups increased with the increase in sample size. For example, Group # 1 had 0.022 while group #3 had -0.128. The strength of relationship between the two variables decreases with increase in sample size.

## Conclusion

Based on the results obtained above, it can be concluded that a change in the sample size has a significant effect on the variability of data. Therefore, it is good to choose a larger sample size in order to correctly obtain good results for in data analysis.

## References

Ramseh, G. (2009). Introduction to confidence intervals. Web.

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