Andy Vetter, Brian Michitti

ECON 257K

April 11, 2005

Hanover Student Engagement vs. Other Public and Private Institutes

A Statistical Analysis

Introduction

Nearly every weeknight Hanover College’s Duggan Library is crammed full of hard working students. It is well known amongst Hanover students that many hours of work outside the classroom are required to have success in the difficult classes of this private college. Rumors constantly fly about the speculation that larger state universities do not require the same amount of effort to have similar success in the classroom. Is there any truth to this belief? Is it possible that Hanover College requires even more work than other private institutes? Are Hanover students able to comprehend information better than students at other institutes? By analyzing different schools mean G.P.A., surveys listing different hours devoted to studying, and entering freshman’s average SAT scores, we plan to answer these questions.

The Data

The data for Hanover students was collected through a randomly distributed survey. 145 students completed this survey, and our project requires the responses for “study”, “G.P.A.” and, “SAT”,. Responses for “study” indicate the amount of hours in a typical week the student spends studying outside of class. On the survey, 7 students left the question “HC GPA” incomplete. 25 students left the question “SAT” incomplete. The reason for the large number of incomplete “SAT” scores on the survey is most likely due to the fact that most states do not required high school students to take both the SAT and the ACT. Therefore many Hanover students may have taken the ACT rather than the SAT, therefore leaving the question unanswered. Only one survey did not have a response for “study”. During analysis of these variables, all empty cells will not be included.

Data gathered from various outside sources will be used to analyze all other institutes. Mean grade point averages and SAT scores were found from “The Princeton Review” website and also the book “American Universities and Colleges”. All of the average study times were found in a survey conducted by “U.S. News” in 2003. The information given by this survey, however, was asked differently than the Hanover survey. Therefore we were required to use weighted averages to make the given percentages compatible with the Hanover survey’s “number of hours per week”.

Summary Statistics

The summary statistics (see table 1) indicate that the mean state university GPA is 3.3025, with a standard deviation f 0.2031. We will be comparing this GPA of 3.3025 to the mean Hanover College GPA of 2.891. The mean SAT scores of incoming state university students is 1081.92. The standard deviation of these SAT scores are approximately 128 points. These mean state university SAT score will also be compared to the mean Hanover SAT (1167.117). The average state university student studies 13.5 hours per week. This will be analyzed with Hanover Students’ mean study time of 17.54 hours per week. Twelve different universities were sampled to form these point indicators.

Table 1: Summary Statistics

	State GPA	State SAT	State Study	Private GPA	Private SAT	Private Study
Mean	3.3025	1081.9166	13.50333	3.225	1198.92	15.7107
Median	3.355	1087	13.425	3.29	1204	15.72
Max	3.52	1294	17.64	3.4	1319	19.45
Min	2.9	921	10.21	2.82	1126	12.04
Std. Dev	0.2031	128.0308	1.814363	0.1725	53.226	2.11052
Count	12	12	12	13	13	13

When looking at the private college statistics, we see that the mean GPA is 3.225, with a standard deviation of 0.1725. Incoming freshman to these private colleges had a mean SAT score of 1198.92. Standard deviation for the mean SAT score was 53.2 points. The mean number of hours study per week by students in private institutes is 15.71, with a standard deviation of 2.11. These statistics will also be analyzed with the mean Hanover College scores and hours listed above. Thirteen private colleges were sampled to form these point indicators.

Do Hanover students study more than state university students?

One may believe that we can answer this question by simply looking at the sample means (see graph 1) of study hours and conclude that because Hanover has a greater mean, they must study more than state school students. This simply isn’t true.

Graph 1: Mean Study Hours

Because this is merely a sample, we do not know what the actual mean study hours for either group actually is. Therefore, we must do some further analysis. Table 2 summarizes the difference in mean number of hours studied per week by Hanover and state university students.

Table 2: Study Hours by Institute (Hanover, State)

	Hanover College	State Universities
Sample Mean	17.54166667	13.50333333
Sample Std. Deviation	10.35041636	1.814363567
Sample Size	144	12

A one-tailed hypothesis test would be required to determine whether the difference between mean hours studied is significantly greater by Hanover students. Because one of our sample sizes is less than 30 (state n=12), we will use the t-distribution test to test our hypothesis:

Ho: m_h - m_s £ 0

Ha: m_h - m_s > 0

At the 90% level of confidence, and 154 degrees of freedom, the critical one tailed “t-value” is 1.2855. Because the test statistic we found for the above data is 1.645, we can reject our null hypothesis. Our test statistics of 1.645 is greater than our critical value of 1.2855, allowing the figure to fall in the rejection range for the null hypothesis. In other words, we are 90% confident that there is a significant difference in the amount of hours studied by Hanover students in comparison to state university students.

Do Hanover students study more than other private college students?

We can once again see that the mean number of hours studied per week by Hanover student is greater than the mean number for the private school students surveyed (see table 3). Once again, because these are samples, and the true mean is unknown, further analysis must once again be done to determine whether or not there is a significant difference between the two figures.

Table 3: Study Hours by Institute (Hanover, Private)

	Hanover College	Private Colleges
Sample Mean	17.54166667	15.71076923
Sample Std. Deviation	10.35041636	2.110527033
Sample Size	144	13

Once again, because we have one sample size less than 30 (private n=13), we must use a t-distribution test. The test will be one-tailed because we wish to see if Hanover’s mean study hours are significantly greater than the mean of other private colleges.

Ho: m_h - m_p £ 0

Ha: m_h - m_p > 0

At the 90% confidence level we are unable to reject the null hypothesis. Our test statistic of .6325 is not greater than our critical value of 1.2855. Therefore we are unable to reject the null hypothesis, and we are unable to conclude that there is a significant difference between Hanover and other private college’s mean study hours.

Do Hanover freshman have higher SAT scores than state university and/or private college freshman?

Now that we have analyzed the amount of time the average student of each institute spends studying, it is important for us now to examine how well the average student tested on the SAT. This could be a strong indicator for the actual learning ability of the average student entering the different institutes. Table 4 summarizes the SAT scores for Hanover freshman in comparison to state university freshman and other private college freshman.

Table 4: SAT of Incoming Freshman (Hanover, State, Private)

	Hanover College	State Universities	Private Schools
Sample Mean	1167.116667	1081.916667	1198.923077
Sample Std. Deviation	126.913865	128.0308616	53.22665613
Sample Size	120	12	13

Because we wish to find if there is any difference between the two sample means, we will be using a two-tailed test. Once again, we have samples of less than 30 (state n=12 and private n=30), we will be using a t-distribution test.

Ho: m_h - m_s/p = 0

Ha: m_h - m_s/p ≠ 0

We find that both tests will have approximately the same critical value of 1.645. This is the critical value for testing at a 90% confidence interval. When analyzing Hanover freshman SAT scores in comparison with state university freshman SAT scores, we find a test statistic of 2.2156. This allows us to reject the null hypothesis, or conclude that there is a significant difference between state and Hanover incoming SAT scores.

Upon comparison of Hanover incoming freshman SAT scores and other private college freshman scores, we find a test statistic of -.89266. Because this does not fall within the rejection range of the null hypothesis (-1.645), we must conclude that there is no significant difference in scores.

How does Hanover’s mean GPA compare to state and other private college GPAs?

The final necessary step in our study will be to compare the mean Hanover GPA to both other private college’s mean GPA and the mean state university GPA. The table below summarizes the data.

Table 5: GPA Summary (Hanover, State, Private)

	Hanover College	State Universities	Private Colleges
Sample Mean	2.891492754	3.3025	3.225384615
Sample Std. Deviation	0.515280892	0.203117745	0.172511612
Sample Size	138	12	13

The two research questions we would like to answer are “Is the mean GPA for other private colleges/ state universities greater than Hanover?” Therefore, the null and alternative hypotheses will appear as follows:

Ho: m_s/p - m_h £ 0

Ha: m_s/p - m_h > 0

A critical value of -2.342 (99% confidence), with 148 degrees of freedom, will be used for first analyzing the GPA of the mean state university GPA and the mean Hanover GPA. We found our test statistic to be -2.738, which falls within our rejection range for our null hypothesis. Therefore, with 99% confidence, we are able to reject our null hypothesis. In other words, we are 99% sure that the mean state university student’s GPA is significantly greater than that of the average Hanover student.

Upon comparison of Hanover’s mean GPA to other private college GPAs, we find a test statistic of -2.553. Our critical value is approximately the same as the value for state universities (only 1 additional degree of freedom). Therefore we are also able to reject the null hypothesis in this test at a 99% confidence level. Thus, we can conclude that the other private institutes sampled have a significantly greater mean GPA than that of the Hanover student mean.

Significance of Findings

After running this series of hypothesis tests, we are able to make some valuable assumptions concerning our questions we originally hoped to be able to answer. When first looking at hours of time spent studying, we find that we are highly confident (90%) that the average student does indeed study more than the typical state university student. However we cannot conclude that there is a significant difference amongst study hours between Hanover and other private colleges. Although this may be something that could be guessed prior to even doing any research or running any tests, when we combine these findings with our other results, we do find some interesting outcomes.

We find no significant difference between the incoming freshmen SAT scores of Hanover freshman and other private college freshman. This allows us to make the assumption that all of these freshmen have approximately the same ability to learn and comprehend information. As previously state, we also find that there is no significant difference in time spent studying. However, when looking at mean GPAs, we find Hanover to have a significantly lower mean GPA than that of other private colleges. Why do students who study approximately the same amount of time with the same ability to comprehend information have different GPAs? We do not know the answer to this question but can make additional hypotheses such as less grade inflation at Hanover, tougher classes, etc. We found it interesting to be able to draw possible conclusions such as this after analysis of our data.

As far as state universities are concerned, we can conclude that the samples have a significantly lower incoming freshman SAT score, lower average number of hours studied per week, but still have a significantly higher mean GPA than that of Hanover students. Again, we did not run tests to figure out why this is the case, but we do feel that this makes economic sense. Private institutes, such as Hanover, on average are more expensive than state schools. Therefore, we could expect these classes to be more challenging, resulting in lower grade point averages.

Conclusion

The reason for us initially deciding to look into the topic that we choose was because we hoped to be able to eventually draw conclusions and form hypothesis similar to what we have done about the relationship between Hanover and other institutes. We found it very interesting for us to actually be able to form some statistically supported answers to questions the average Hanover student frequently wonders.

Looking at our findings from an economical perspective, we believe it all makes sense. It can be concluded that private school students must devote more time to their studies than state university students, only to see grades that are on average significantly lower. Because these private institutes generally cost more than large state universities, we should also expect the challenge to be greater. We believe it would probably be a disappointment to the person paying the tuition if this was not the case. The existence of this challenge is represented by the results of our data.

One hundred and forty-five Hanover students, 12 randomly selected state universities, and 13 randomly selected private colleges made up our different categories for comparison. We believe putting Hanover in comparison with individually selected institutes (Harvard, Yale, etc.) would also be interesting for analysis. However, we feel that the results presented above are able to provide for a good understanding of how Hanover College scholastically compares to other public and private institutes.