Correlation and Chi-Square Tests in Statistics

This document discusses correlation and chi-square tests in statistics. It explains the concepts, hypotheses, and types of tests. It provides examples and explanations for better understanding.

Statistics
Name:
Institution:
20th March 2019

Part 1:
Q1:
Correlation refers to a statistical technique which shows how a pair of variables are related
(strong or weak) and the direction of change in one variable as the other changes (Nikolić,
Muresan, Feng, & Singer, 2012). Causation on the other hand, indicates that one event is the result
of the occurrence of the other event; i.e. there is a causal relationship between the two events.
Example is a case of time students take to prepare for exams and the student’s score in exam.
There is a positive correlation between the time students take to prepare for exams and the
student’s score in exam. However, time students take to prepare for exams does not cause the
student’s score in exam.
Q2:
The hypothesis is;
Null hypothesis (H0): There is no significant correlation between a player’s 40 yard dash time
and height of player’s vertical leap.
Alternative hypothesis (HA): There is significant correlation between a player’s 40 yard dash
time and height of player’s vertical leap.
The suggested type of test is the Pearson r correlation because the variables are thought to be
linearly related (Mahdavi Damghani, 2012).
The value of the Pearson correlation shows that a weak negative relationship exists between
Player’s 40 yard dash time and the height of the player’s vertical leap. The critical value is 0.444

at α = 0.05 significance level. Thus, an observed r = -.29 in a sample of n=20 observations is not
significantly different from random.
The coefficient of determination is 0.0841; this means that 8.41% of the variation in the
dependent variable (Player’s 40 yard dash time) is explained by the height of the player’s vertical
leap.
Q3:
The hypothesis is;
Null hypothesis (H0): There is no significant correlation between student motivation levels and
the student’s GPA.
Alternative hypothesis (HA): There is significant correlation between student motivation levels
and the student’s GPA.
The suggested type of test is the Pearson r correlation because the variables are thought to be
linearly related.
The results of the Pearson r correlation are given in table below;
Table 1: Correlations
Motivation GPA
Motivation Pearson Correlation 1 .434*
Sig. (2-tailed) .017
N 30 30
GPA Pearson Correlation .434* 1
Sig. (2-tailed) .017
N 30 30
*. Correlation is significant at the 0.05 level (2-tailed).
The results shows that a moderate and significant positive linear relationship exists between
motivation level and the GPA score (r = 0.434, p = .017).

The coefficient of determination can be computed as follows;
Coefficientofdetermin ation=r2=0.4342=0.188356
The coefficient of determination is 0.1884; this means that 18.84% of the variation in the
dependent variable (GPA score) is explained by the motivation level.
Part 2:
Q1:
The hypothesis to be tested is;
Null hypothesis (H0): People prefer all the candy bars equally.
Alternative hypothesis (HA): People don’t prefer all the candy bars equally.
The suggested type of test is the Chi-Square test since there is only categorical variables
involved in this study. Chi-square test is used to test for relationships that exists between
categorical variables (Bagdonavicius & Nikulin, 2011).
Q2:
The hypothesis to be tested is;
Null hypothesis (H0): There is no significant association between gender and offense type.
Alternative hypothesis (HA): There is significant association between gender and offense type.
The suggested type of test is the Chi-Square test since we need to test for the relationships
between two categorical variables.
We are given data as follows;

Table 2: gender versus offense type
Males Females Total
Violent 30 12 42
Non-violent 250 200 450
Total 280 212 492
We now compute the Chi-Square value as follows;
χ2=∑ ( Oi −Ei )2
Ei
E1= 42∗280
492 =23.902
E2= 450∗280
492 =256.098
E3 = 42∗212
492 =18.098
E4= 450∗212
492 =193.902
So we have;
O1=30 , O2=250 , O3=12 ,O4=200 , E1=23.902 , E2=256.098 , E3=18.098 , E4=193.902
χ2=∑ ( Oi −Ei )2
Ei
Substituting these values to the above formula we have;
( 30−23.902 ) 2
23.902 + ( 250−256.098 ) 2
256.098 + ( 12−18.098 ) 2
18.098 + ( 200−193.902 ) 2
193.902 =1.5555+ 0.14518+2.054434+0.191747

3.986862
The critical chi-square value was obtained as 3.8415 ( χ2 ( 1 )=3.8415)
Decision
We reject the null hypothesis since the computed chi-square value is greater than the critical chi-
square value.
Conclusion
We conclude that there is significant association between gender and offense type. More males
are involved in violent crime as compared to the females.
Q3:
The hypothesis to be tested is;
Null hypothesis (H0): There is no significant association between type of offense and the type of
sentence.
Alternative hypothesis (HA): There is significant association between type of offense and the type
of sentence.
The suggested type of test is the Chi-Square test since we need to test for the relationships
between two categorical variables.
Q4:
The hypothesis to be tested is;
Null hypothesis (H0): People prefer different kinds of beer equally.

Alternative hypothesis (HA): People do not prefer different kinds of beer equally.
The suggested type of test is the Chi-Square since there is only categorical variables involved in
this study. Chi-square test is used to test for relationships that exists between categorical
variables (Bagdonavicius & Nikulin, 2011).

References
Bagdonavicius, V., & Nikulin, M. S. (2011). Chi-squared goodness-of-fit test for right censored
data. The International Journal of Applied Mathematics and Statistics, 8(4), 30–50.
Mahdavi Damghani, B. (2012). The Misleading Value of Measured Correlation. Wilmott, 1(1),
64–73. doi:10.1002/wilm.10167
Nikolić, D., Muresan, R. C., Feng, W., & Singer, W. (2012). Scaled correlation analysis: a better
way to compute a cross-correlogram. European Journal of Neuroscience, 35(5), 1–21.
doi:10.1111/j.1460-9568.2011.07987.x