Statistics 6: Hypothesis Testing Assignment - Analysis of Variables

This statistics assignment solution explores hypothesis testing and correlation analysis between various variables, including incarceration rates, median state income, percentage of adults with post-secondary degrees, and total crime rates. The solution begins by formulating null and research hypotheses for the relationships between these variables. It then delves into the concept of the correlation coefficient, explaining its significance and how it quantifies the degree of association between variables. The document predicts the strength and direction of these relationships, arguing that the association between incarceration rate and total crime rate would be the strongest and positive. Finally, it discusses the interpretation of scatterplots and the implications of the absence of a straight-line relationship, clarifying that it doesn't necessarily mean no relationship exists. The solution references relevant statistical concepts and definitions, along with the expectation that the degree of association between the incarceration rate and the median state income to be positive but weaker than the degree of association between incarceration rate and median state income. The solution concludes with a discussion on scatter plots and the interpretation of correlation.

Running head: STATISTICS 1
Statistics: Hypothesis Testing
Student Name
Institution

STATISTICS 2
Question 1
Null hypothesis: Incarceration rate and median state income are not significantly related
Research hypothesis: A significant relationship between the median state income and
incarceration rate exists
Question 2
Null hypothesis: No significant relationship between incarceration rate and the percentage of
adults with post-secondary degrees
Research hypothesis: A significant relationship exists between incarceration rate and the
percentage of adults with post-secondary degrees
Question 3
Null hypothesis: Incarceration rate and the total crime rate are not significantly related
Research hypothesis: A significant relationship between incarceration rate and total crime rate
exists
Question 4
In mathematics and statistics, the degree or the magnitude of association between two
variables thought to be related is quantified by calculation of correlation coefficient. A negative
value of correlation coefficient suggests that two variables are negatively associated such that a
unit increase in one variable results to a decrease in the other unit. Conversely, a correlation
coefficient greater than zero implies that two variables are positively associated such that a unit
increase in one variable results to an increase in the other unit. The correlation coefficient value
ranges from -1 to 1 (Behara & BarCharts, 2014). The mathematical symbol for the correlation
coefficient is denoted by r. Strong positive linear relationships have a correlation coefficient
value of between 0.8 and 1 (Davies, 2017).

STATISTICS 3
Moreover, a weak positive linear relationship exists between two values if the correlation
coefficient value lies between 0 and 0.4. Conversely, strong negative linear relationships have a
correlation coefficient value of between -0.8 and -1 (Lockow, 2012; Hartmann, 2016). Moreover,
a weak negative linear relationship exists between two values if the correlation coefficient value
range between 0 and -0.4.
Of the three relationships, I expect the degree of association between incarceration rate
and total crime rate to be the highest. However, the degree of association between incarceration
rate and the total crime rate is expected to be positive as opposed to the degree of association
between incarceration rate and median state income and the degree of association between
incarceration rate and percentage of adults with post-secondary degrees. The total crime rate in
the United States and the incarceration rate is strongly related. Chances of a person engaging in
any criminal activity being taken to prison are higher than a person earning minimum wage
salary. Similarly, chances of a person engaging in any criminal activity being taken to prison are
higher than an adult with post-secondary degrees committing a crime then imprisoned.
Similarly, I expect the degree of association between incarceration rate and the median
state income to be positive but weaker than the degree of association between incarceration rate
and median state income. The probability of an individual earning a median state income would
engage in any criminal activity would be close to zero. Therefore, the degree of association
between the incarceration rate and the median state income is weak.
Lastly, I expect the degree of association between the percentage of adults with post-
secondary degrees to be negative and weaker than the degree of association between
incarceration rate and median state income and the degree of association between incarceration
rate and total crime rate. The probability of adults with post-secondary degrees engaging in any

STATISTICS 4
criminal activity would be close to zero. Therefore, the correlation between incarceration rate
and adults with post-secondary degrees negative and weak.
Question 5
A scatterplot is a visual representation of the relationship between two variables. In
situations where the plotted points lie in a straight line, then a relationship is said to be linear.
However, in the absence of a straight-line relationship between two variables does not mean the
nonexistence of relationship between the variables. Relationship between variables could be non-
linear or cyclical. For instance, in time series plot, a straight-line relationship does not exist, but a
cyclical relationship exists. Such cyclical relationships predict trends and movement of the
variables in question with time.
Absence of a straight-line relationship between two variables could signify a weak linear
relationship. For instance, when the distance of the individual points is very big enough, then a
straight line relationship might be absent.
In contrast, the absence of a straight-line relationship between the two variables could
also mean no relationship between two variable. For instance, if the correlation coefficient, r is
zero, then a straight-line relationship between two variables does not exist. Consequently, no
significant relationship between the two variables.

STATISTICS 5
References
Behara, R. and BarCharts, I. (2014).Business Statistics. [Boca Raton, FL]: QuickStudy Reference
Guides (Quick Study Business). Available at: http://search.ebscohost.com/login.aspx?
direct=true&db=nlebk&AN=1534295&site=ehost-live (Accessed: 9 August 2019).
Davies, A. (2017). Understanding Statistics : An Introduction, Libertarianism.org Press,
Washington, D.C., viewed 9 August 2019, <http://search.ebscohost.com/login.aspx?
direct=true&db=nlebk&AN=1667844&site=ehost-live>.
Hartmann, H. (2016).‘Statistics for Engineers’, Communications of the ACM, 59(7), pp. 58–66.
doi: 10.1145/2890780.
Lockow, E. (2012) ‘S. J. Morrison: Statistics for Engineers: an introduction’, Statistical Papers,
53(1), pp. 245–246. doi: 10.1007/s00362-010-0358-x.