Part 1:
The null hypothesis states, there is no linear association
between the crime rate and the percentage of free lunches given out. The
alternative hypothesis states, that there is a linear relationship between
crime rate and the percentage of free lunches given out. In order to determine
whether or not a linear relationship exists between the two variables of crime
late and percent free lunches, linear regression analysis is used. The data for
each variable was ran through a linear regression analysis using SPSS, with a
significance level of 0.05. The results of the analysis provide an equation of
the two variables of Y=21.819 + 1.685X where crime rate is the dependent
variable represented by Y, and percent free lunch is the independent variable,
represented by X. The equation indicates a positive linear correlation
explaining that for every one increase in percentage of free lunches, the crime
rate in turn increases by 1.685. Therefore predictions can be made using the
equation. For example, approximately 34.35 free lunches are given out in areas
with a crime rate of 79.7. Although the equation and scatter plot represent a
distinguishable positive correlation, the actual linear relationship between
the variables is extremely weak as indicated by an R2 value of only
0.173, which also makes predictions less accurate. However, based on the
results of the regression analysis, the null hypothesis can be rejected because
of a significance level less than 0.05, at 0.005. Thus, there is a weak linear
association between crime rate and the percent of free lunches given out.
Part 2:
Introduction:
The UW system is curious in determining whether or not
certain factors influence the amount of enrollment at two different schools,
the University of Wisconsin Milwaukee and the University of Wisconsin Eau
Claire. The amount of enrollment at each university may be influenced by
factors such as the amount of income and education in certain counties, as well
as the distance of each county away from each university. These variables can
determine a student’s decision in deciding between different universities, thus
affecting the amount of enrollment at different schools. Data regarding the
enrollment amount at each university as well as income, percent bachelor’s
degree, and distance for each county in Wisconsin is used. In order to
determine whether or not these variables influence the amount of enrollment the
data is analyzed using regression analysis. After performing regression
analysis on the data the UW System can determine which factors are most
significant in influencing enrollment amounts at the University of Wisconsin
Milwaukee and the University Wisconsin Eau Claire. When significant factors are
determined, spatial representations are used in relation to the regression
statistics to determine spatial patterns of enrollment based on the most influential
variables.
Methodology:
Regression analysis will determine whether or not to reject
the null hypothesis, stating that there is no relationship between each
variable and enrollment at both universities. If statistically significant,
then the alternative hypothesis, stating there is a linear relationship between
each variable and enrollment at both universities, can be investigated. In
order to properly determine which variables have the most significant influence
on the amount of enrollment at each university the data is analyzed through
regression analysis in SPSS. Regression
analysis statistics are performed in SPSS to determine whether or not any of
the three suspected variables have a significant relationship to the amount of
enrollment at each university. Six different regression analyses are performed
using the enrollment data for both universities in relation to each of the
three variables. The results each analysis will indicate which variables have
significant relationships to the amount of enrollment at each university, thus
indicating which factors are more influential on a student’s decision to
attended different universities.
The data for the three variables include median household
income in each county, percent bachelor’s degrees in each county, and the
distance of each county (from its center) away from each university. The data
regarding median household income and percent bachelor’s degree for each county
are ready for analysis and do not need to be normalized or altered. However, the
distance data must be normalized based on the population for a more accurate
analysis of the data. This is done by dividing the distance for each county by
the population for each county. The normalized distance data will be used in
the regression analysis.
Three separate regression analyses are performed using the
enrollment data for the University of Wisconsin Eau Claire in relation to each
of the three variables. The regression analysis will indicate whether or not
there is a relationship between each variable and the enrollment amount at Eau
Claire as well as create an equation of the relationship providing a means to
make predications regarding that relationship. The goal is to establish whether or not the
enrollment amount at the university depends on variables such as income,
percent bachelor’s degrees, and distance. Because of this the enrollment data
for this university is used consistently as the dependent variable in order to
determine how the other variables, the independent variables, influence enrollment
amount. After three separate regression analyses were performed comparing the
University of Wisconsin Eau Claire enrollment data to the suspected independent
variables, the same was done using the University of Milwaukee enrollment data
compared to the same three independent variables. The results of each
regression analyses in SPSS will indicate which variables have a significant
relationship with the enrollment amounts at each university as well as the
pattern of that relationship in the form of an equation.
After the regression analysis provides the statistics
determining the most significant variables, the data for those variables can be
graphed in relation to the enrollment data for each university. The graphs will
provide a visual interpretation of the trends associated with each significant
variable. A scatterplot will display the actual pattern of the raw data in
comparison to a trend line with an equation determined by regression analysis.
The observed data plotted in comparison to a trend line representing the
predicted relationship helps to visually identify both the pattern and strength
associated with the relationship of a given variable with the amount of
enrollment at both universities.
In order to better understand the most influential factors,
spatial representations of each significant variable are produced to be
examined in relation to the regression statistics. The spatial representations
map the residuals of the statistically significant variables. The residuals
indicate the amount the actual data deviates from the predicted value of the
relationship provided by the equation. Residuals that are closer to zero
indicate no deviation of the actual data to the predicted outcome, meaning a
relationship between variables can be accurately predicted. The further the
residual is from zero in either direction indicates a less accurate prediction
of a relationship. The residuals for each county for the significant variables
can be saved in SPSS during regression analysis to be used in ArcMap. The maps
created in ArcMap of the residuals help identify which counties in Wisconsin
are accurate representations of specific factors influencing enrollment at each
university, and which counties appear as outliers. Establishing areas where
outliers are occurring allows for a clearer interpretation of certain patterns
regarding the influence of specific factors on enrollment at each
university.
Results:Based on the results of the regression analyses comparing the Enrollment amount at the University Wisconsin Eau Claire and each of the three variables, two of the three variables were found to be statistically significant. The null hypothesis is rejected regarding both percent bachelor degrees and the distance variable. Therefore, there is a significant linear association between percent bachelor’s degrees and Eau Claire enrollment as well a significant relationship between distance and enrollment. The null hypothesis is rejected for both these variables considering the regression analysis provided statistics with a significance level below 0.05. However, the variable regarding income did not show a significant association to enrollment at Eau Claire after regression analysis. A significance level for this variable greater than 0.05 fails to reject the null hypothesis, meaning there is not a significant linear relationship between median house hold income and Eau Claire enrollment.
After establishing which variables are significant in
regards to influencing enrollment at Eau Claire, further analysis of the
regression statistics provides information about the strength and direction of
that relationship. In regards to the influence of percent bachelor’s degrees on
Eau Claire enrollment, the relationship provided by an equation of
Y=-126.472+4283.038X and an R2 of 0.121 indicates a weak but
apparent positive linear association. Thus, for every one increase in
percentage of bachelor’s degrees enrollment at Eau Claire increases by approximately
43 students. However, even though the relationship is proved to be statistically
significant the predictions that can be made provided by this equation are
fairly inaccurate considering the R2 of 0.121 is relatively low, representing
a weak relationship. On the other hand, the distance variable not only proves
to be statistically significant, but the relationship between distance and
enrollment is much stronger. The equation of Y=8.518+0.124X and an R2
value of 0.945 indicates a strong positive relationship between the two
variables. As the distance increases, enrollment in turn increases at a rate of
0.124. Where counties 500 miles away have a typical enrollment of 70 students
at Eau Claire. Predications that are made using the equation for this variable
are fairly accurate considering the strong relationship provided by the R2
value of almost 0.945, but the rate of increase is fairly minimal.
Based on the results of the regression analyses comparing
enrollment amounts at the University of Wisconsin Milwaukee and each of the
variables, all three of the variables were proved to be statistically
significant. Because of a given significance level of less than 0.05 in all the
results we can reject the null hypothesis concerning all three variables. Therefore,
there is linear association between Milwaukee enrollment and distance, as well
as linear relationship between Milwaukee enrollment and both percent bachelor’s
degrees and income. Despite the relevance of all three variables being
statistically significant, the strength and pattern concerning each
relationship still needs to be examined.
The relationship between median house hold income and
Milwaukee enrollment is the weakest of the three relationships. The equation of
Y=-1006.75+0.039X created through regression analysis displays a positive
linear relationship between income and enrollment. For every increase in median
house hold income, enrollment increases by 0.039. A median household income of
around 30,000 for a county contributes to about 164 students at Milwaukee. Despite
the ability to make predictions of the influence of income on enrollment using
the equation, the R2 value of 0.068 indicates a very weak
relationship between the variables making predictions less accurate. The
relationship concerning the influence of percent bachelor’s degrees on
Milwaukee enrollment shows a slightly stronger relationship. Although the relationship
between the variables is slightly stronger than the last, it is still fairly
weak as indicated by an R2 value of 0.16. The equation of
Y=-1082.762+24556.66X explains a positive linear relationship, where for every
one percentage increase in the amount of bachelor’s degrees, enrollment
increases by about 245 students. Predictions made from this equation will
likely be inaccurate concerning the weak relationship between the variables. However,
the relationship regarding the influence of distance on enrollment at Milwaukee
has a much stronger linear association. The R2 value of 0.922 identifies
a strong association between the variables and the equation of Y=108.041+0.015X
shows a positive relationship. For counties 500 miles away there is an
enrollment of 115 student and is increasing by 0.015 students per mile away. Predictions
concerning the influence of distance on Milwaukee enrollment are fairly
accurate taking into consideration the overall strength of the relationship, however
the rate of increase of enrollment per increase in distance in minimal.
Further connections can be made when results of the
regression analyses are considered in relation to the spatial representations
of the residuals for the significant variables. The areas on the maps that
display residuals further away from zero can be determined as outliers, meaning
those are areas that do not follow the expected prediction given by the
equations.
The maps on the bottom are displays the residuals of the
relationship between the University of Wisconsin Milwaukee and each of the
three significant factors. The map on the bottom right shows the residuals of
the relationship between percent bachelor’s degrees and enrollment. In this
map, the residuals in green are the ones closest to zero making those counties
ones that are accurate predictions of the influence of bachelor degrees. The
light blue counties, like the green, are also fairly accurate. However, the
counties in dark blue and yellow are the counties where the percent bachelor’s
degrees do not provide an accurate representation of the enrollment at
Milwaukee, and that the influence of bachelor’s degrees in these areas is not
as strong. The map in the center portrays the residuals of the relationship
between distance and Milwaukee enrollment. Majority of the state, shown in
yellow, follow the predicted pattern of the influence of distance on enrollment
considering those are the counties with residuals closet to zero. There are a
select few counties, particularly the ones in blue, with residuals much higher
than zero, meaning distance in not a significant influence in those areas and cannot
be used to accurately determine enrollment. The last map shows the residuals
for the relationship between income and enrollment. Much of the map, in green
and some yellow, indicate that income is a predictable factor of influence for
determining enrollment in those counties. There are, however, a couple of
outliers in blue indicating income is not an influential factor contributing to
Milwaukee enrollment.
Conclusion: When considering the statistics as well as the residual maps conclusions can be made about influential factors determining enrollment at different schools. Not only can the statistics determine which factors are statistically significant and have the most influence but they also provide information concerning the pattern and strength of the influence. This information is particularly helpful when used in relation to the residual maps, as certain significant factors of influence vary based on location. Overall, the statistics can provide the means to determine which factors are most influential, but the maps allow clearer interpretation of where each variable has the most influence. Some factors deemed the most influential in determining enrollment at different schools are more significant in some counties compared to others. Because of this different areas seem to be more influenced by one variable, and may not be as influenced by another.
The most significant factors influencing enrollment at the University
of Eau Claire include the percentage of bachelor’s degrees in each county as
well the distance away from the university. Even though both of these factors
have a statistically significant relationship with the amount of enrollment,
the influence varies on a county level. When considering the influence of the
percent bachelor’s degrees it is clear that much of the state follows the
predicted pattern associated with the relationship to enrollment. Thus, much of
these areas indicate that the amount of bachelor’s degrees has a predictable
influence on enrollment. However, certain areas in the center of the state
along with a few counties to the north do not follow the predicated
relationship between bachelor degrees and enrollment. Because of this the
influence of the amount of bachelor’s degrees on enrollment is not as strong in
these areas. In contrast, the same counties, along with a few others, are clearly
influenced by distance and have a strong connection to the pattern associated
with the relationship between distance and enrollment. Coincidentally much of
the counties that follow this pattern are near the University of Wisconsin Eau Claire,
therefore it is not surprising to conclude distance is an influential factor in
these areas.
The most significant factors which influence the enrollment at the University of Wisconsin Milwaukee include, percentage of bachelor’s degrees in each county, the median income in each county, as well as distance away from the university. Much of the state is equally influenced by percentage of bachelor’s degrees in the sense that most counties follow the pattern associate with the relationship, as shown by the residual map. However, Milwaukee County does not follow the same pattern as the rest of the state, where the influence of bachelor’s degrees on enrollment is minimal, and that other factors have much more influence in this county. Similar the lack of influence associated with bachelor’s degrees on enrollment in Milwaukee County, income is another factor that is not as influential. Much of the rest of the state has a predictable enrollment amount associated with the influence of income, however Milwaukee County does not. The influence of distance in Milwaukee County, on the other hand, appears to be the most predictable influence on enrollment.
Based on analysis of all the data, it is easy to determine
that the most significant factor influencing enrollment at both university is
distance. When considering how other significant factors influence enrollment
at each university, the influence is not the same throughout the state. While
some counties may be more influenced by the percentage of bachelor’s degrees
other counties, specifically the ones closer to the university, are more
influenced by distance.
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