Statistics Case

In this comparative study, I try to statistically compare and analyze different states that require a competency test for graduation versus states that do not require this. In this study, I will examine whether there are any significant differences in the variables like: SAT scores, Expend, Takers, Enroll, Teacher, Ratio, and the Salary of states based on type of state. It was very interested to see how the descriptive statistics measures of the variables like SAT scores, Expend, Takers, Enroll, Teacher, Ratio, and Salary differ between the states that require a competency test versus the states that do not require the test. A summary table of these descriptive statistics differences is provided in Appendix B.

From the table, you can see that the SAT score for the states that do require the competency test has a mean of 944.65. That mean is based on 17 states. Now, the 34 states that do not require the competency test have a mean of 973.35. As you can see, both of these average SAT scores do not exceed 1000. The contingency table provided in Appendix B give a better break down of the how many states actually exceed 1000. Now, when you look at the median SAT score for the states that require the competency test is 909 and 961 for the states that do not require the competency test. You can also see from the table that the standard deviation is pretty close for each sample too. From looking at the tables in Appendix B, you can see that the mean and median for the SAT scores are higher in the states that do not require the competency test.

The current expenditure for public elementary and secondary education for the 1995-96 school years, in billions of dollars is indicated by the variable Expend. From the tables in Appendix B, you can see that the mean for the Expend in states that do require the competency test is $7.21 billion and the $3.93 billion for the states that do not require the competency test. When you look at the median for the Expend, the difference wasn't that big. It was $3.499 billion for states that require the competency test and $2.878 billion for states that do not require the competency test. What this is probably saying this that there are some high expenditures that are having an effect on the mean.

As you take a further look at the tables in Appendix B, you see that the mean is at 37% for the Takers in the states that required the competency test and 34.88% in the states that did not require the competency test. But the median shows a bigger difference here though with 45% for states that requires the test and 25% for states that don't require it. The mean of the Enroll was at 1268.30 for states that require the competency test compared to the 679.40 for the states that do not require the test. Now the mean for the Teachers was 72.88 for states that require the competency test and 39.63 for states that do not. That is a pretty significant difference. Another important thing to compare is the Ratio. When you look at the tables, you can see that in states that require the competency test, the ratio is 16.92 and 16.69 in states that do not require it. You can also note that the median was not that different either. What this shows is that the student to teacher ratio is pretty equal in these types of schools. And the salary showed only a slight difference when compared. The states that required the test was $34.68 thousand compared to the $36.71 thousand for the states that did not require the testing.

Next I tested some hypothesis. The first hypothesis tested is whether there is a significant difference in the mean of the SAT scores between the states that require the competency test and the states that do not require the competency test. When testing this hypothesis it was concluded that the null hypothesis H0 is that the mean of the SAT score of states that require competency tests is equal to the mean of the SAT score of states that do not require competency tests. And the alternative hypothesis H1 is that the mean of the SAT score of states that require competency tests is not equal to the mean of the SAT score of states that do not require competency tests. To test this hypothesis the significance level α =0.05. Since the standard deviations are unknown we use a t test statistic assuming that the distribution of SAT is normal. The test statistic used to test the hypothesis is t = ((X ̅_1-X ̅_2 ))/√((S_1^2)/n_1 +(S_2^2)/n_2 ) which follows a student t distribution with 32 df (Key, 1997). The detailed results of the hypothesis are shown in the table under Appendix C labeled SAT (Hypothesis 1). From the table you can see that the value of the test statistic is t = -1.2898 and the two tailed p value is 0.2064. Since, the p value is larger than the significance level; we fail to reject the null hypothesis at 5% level of significance. Therefore, it can be conclude that there is not enough evidence to conclude that the mean of the SAT score of states that require competency tests is different from the mean of the SAT score of states that do not require the testing.

Next I tested whether there is a significant difference in means current expenditure for public elementary and secondary education between the states that require competency tests and the states that do not require competency tests. The test is conducted that the null hypothesis H0 is that the mean of the expenditure of states that require competency tests is equal to the mean of the expenditure of states that do not require competency tests. And the alternative hypothesis H1 is that the mean of the expenditure of states that require competency tests is not equal to the mean of the expenditure of states that do not require competency tests. To test this hypothesis the significance level α =0.05 was also used.

Since the population standard deviations are unknown we use a t test statistic assuming that the distribution of expenditure is normal. The test statistic used to test the hypothesis is t = ((X ̅_1-X ̅_2 ))/√((S_1^2)/n_1 +(S_2^2)/n_2 ) which follows a student t distribution with 19 df (Key, 1997). The detailed results of the hypothesis are shown in the table under Appendix C labeled Expend (Hypothesis 2). From the table you can see that the value of the test statistic is t = 1.5524 and the two tailed p value is 0.1371. Since, the p value is larger than the significance level; we fail to reject the null hypothesis at 5% level of significance. Therefore, it can be concluded that there is not enough evidence to conclude that the mean of the expenditure of the states that require competency tests is different from the mean of the expenditure of states that do not require the competency tests.

Next hypothesis tested is whether there is a significant difference in the mean of the percentage of graduates who took the SAT between the states that require competency tests and that do not require competency tests. The