Statistics Assignment: Hypothesis Testing - Problem Set 2, Fall 2024

This statistics assignment presents a comprehensive analysis of a dataset, focusing on hypothesis testing and statistical inference. The solution begins with descriptive statistics, including measures of central tendency and dispersion for numerical variables like age, police satisfaction, social cohesion, and fear. It then proceeds to conduct one-sample t-tests to assess the mean fear score and police satisfaction score, comparing them against specific values. The assignment also involves identifying outliers in the data and constructing confidence intervals for the mean age and social cohesion score. Furthermore, it explores the relationships between categorical variables, employing chi-square tests to examine the independence between race and direct experience with police (XP), speeding and feeling unsafe downtown, police contact and age group, and feeling unsafe downtown and age group. The analysis includes the formulation of hypotheses, selection of appropriate statistical tests, interpretation of p-values, and drawing conclusions about the relationships between variables. The solution provides detailed tables and figures to support the statistical findings and conclusions.

Statistics
Statistics
Student name:
Tutor name:
1 | P a g e

Statistics
Question 1a
Introduction
The sample size involved in this study was of size 50. This sample can be assumed to be
normally distributed since its size is greater than 30. The variables involved in this sample are
both categorical and numerical. The numerical variables are age, police sat, social cohesion and
fear. The categorical variables are direct XP, race, speeding and unsafe DT. All the categorical
variables are dichotomous in nature. Dichotomous variables are those variables that have two
responses. Examples are yes or no, male or female and many others.
Question 1b
Descriptive statistics for numerical variables
Table of results
Statistics
Age Police sat Social
cohesion
Fear
N Valid 50 50 50 50
Missing 0 0 0 0
Mean 44.6000 38.2400 21.3200 11.2600
Median 41.0000 39.0000 21.5000 10.0000
Mode 18.00a 45.00 23.00 9.00a
Std. Deviation 20.927 12.46638 3.99714 4.07486
Variance 437.959 155.411 15.977 16.604
a. Multiple modes exist. The smallest value is shown
Table 1
The table above shows the measures of center and measures of central tendencies for the four
numerical variables. The mean, median and mode for the variable age are 44.6, 41 and 18 years
respectively. The mean, median and mode for the variable police sat are 38.24, 39 and 45. To
2 | P a g e

Statistics
add on, the mean, median and mode for the variable social cohesion are 21.32, 21.5 and 23.
Lastly, the mean, median and mode for the variable fear are 11.26, 10 and 9.
Question 2a
Test for mean of fear
Hypothesis
H0: The mean fear score is equal to 12.5
Versus
H1: The mean fear score is not equal to 12.5
The study will employ a one sample t-test since we are comparing a mean against a single
variable. The alpha value is 0.05
Table of results
One-Sample Test
Test Value = 12.5
t df Sig. (2-tailed) Mean Difference 95% Confidence Interval of the
Difference
Lower Upper
Fear -2.152 49 .036 -1.24000 -2.3981 -.0819
Table 2
The table above shows the results of the one sample t-test. We can observe that the calculated p-
value is 0.04. This is less than the alpha value which is 0.05. The decision therefore is that the
null hypothesis is not accepted. The conclusion therefore is that the mean fear score is not equal
to 12.5. This means that the sample is not a representative of the whole population.
3 | P a g e

Statistics
Question 2b
Test for mean of fear
Hypothesis
H0: The mean score level of police satisfaction score is equal to 33.
Versus
H1: The mean score level of police satisfaction is not equal to 33.
The study will employ a one sample t-test since we are comparing a mean against a single
variable. The alpha value is 0.05
Table of results
One-Sample Test
Test Value = 33
t df Sig. (2-tailed) Mean Difference 95% Confidence Interval of the
Difference
Lower Upper
Police sat 2.972 49 .005 5.24000 1.6971 8.7829
Table 3
The table above shows the results of the one sample t-test. We can observe that the calculated p-
value is 0.01. This is less than the alpha value which is 0.05. The decision therefore is that the
null hypothesis is not accepted. The conclusion therefore is that the mean fear score is not equal
to 33. This means that the sample is not a representative of the whole population.
4 | P a g e

Statistics
Question 3
Test for presence of outliers
Figure 1
As can be observed in figure 1 above, the variables age and police satisfaction have no outliers
while the variables social cohesion and fear had outliers. Social cohesion had fewer outliers on
the extreme lower end. Fear had more outliers but to the extreme upper end.
Question 4a
95% and 99% confidence interval for the mean age
Age
Confidence Level
(95.0%)
5.94752
2
Mean 44.6
Lower bound
38.6524
8
Upper bound
50.5475
2
5 | P a g e

Statistics
Table 4
Age
Confidence Level
(99.0%)
7.93156
4
Mean 44.6
Lower bound
36.6684
4
Upper bound
52.5315
6
Table 5
As can be observed from the two tables above, at 95% confidence interval, the mean age lies
between 38.65 years and 50.54 years at 99% confidence interval, the mean age lies between
36.66 years and 52.53 years. Since the larger population has got a mean of 54 years which is not
covered by the samples intervals, it can be concluded that the sample was not a good
representation of the larger population.
Question 4b
95% and 99% confidence interval for the mean social cohesion score
Social Cohesion
Confidence Level
(99.0%)
1.51492
7
Mean 21.32
Lower bound
19.8050
7
Upper bound
22.8349
3
Table 6
Social Cohesion
Confidence Level 1.13597
6 | P a g e

Statistics
(95.0%) 5
Mean 21.32
Lower bound
20.1840
2
Upper bound
22.4559
8
Table 7
As can be observed from table 6 and table 7 above, at 95% confidence interval, the mean social
cohesion score lied between 20.18 years and 22.46 and at 99% confidence interval, the mean
social cohesion score lied between 19.8 and 22.83. Since the larger population has got a mean of
20 which is covered by the sample’s 99% confidence interval, it can be concluded that the
sample was a good representation of the larger population.
Question 5
Test for independence between race and XP
Hypothesis
H0: Race and XP are independent.
Versus
H1: There is a relationship between race and XP.
The study will employ chi-square since we are testing independence between two variables. .
The alpha value is 0.05
Table of results
Chi-Square Tests
Value df Asymp. Sig. (2-
sided)
Exact Sig. (2-
sided)
Exact Sig. (1-
sided)
Pearson Chi-Square .512a 1 .474
Continuity Correctionb .182 1 .669
7 | P a g e

Statistics
Likelihood Ratio .515 1 .473
Fisher's Exact Test .569 .335
Linear-by-Linear Association .502 1 .479
N of Valid Cases 50
a. 0 cells (0.0%) have expected count less than 5. The minimum expected count is 9.24.
b. Computed only for a 2x2 table
Table 8
The table above shows the results of the chi-square test. We can observe that the calculated p-
value is 0.57. This is greater than the alpha value which is 0.05. The decision therefore is that the
null hypothesis is accepted. The conclusion therefore is that the race and XP are independent.
Question 6
Test for relationship between speeding and unsafe feel
Hypothesis
H0: Thought of speeding and feeling unsafe downtown are independent.
Versus
H1: There is a relationship between the thought of speeding and feeling unsafe downtown.
The study will employ chi-square since we are testing independence between two variables. .
The alpha value is 0.05
Table of results
Chi-Square Tests
Value df Asymp. Sig. (2-
sided)
Exact Sig. (2-
sided)
Exact Sig. (1-
sided)
Pearson Chi-Square .487a 1 .485
Continuity Correctionb .166 1 .684
Likelihood Ratio .486 1 .486
Fisher's Exact Test .567 .342
8 | P a g e

Statistics
Linear-by-Linear Association .477 1 .490
N of Valid Cases 50
a. 0 cells (0.0%) have expected count less than 5. The minimum expected count is 8.80.
b. Computed only for a 2x2 table
Table 9
The table above shows the results of the chi-square test. We can observe that the calculated p-
value is 0.57. This is greater than the alpha value which is 0.05. The decision therefore is that the
null hypothesis is accepted. The conclusion therefore is that there is a relationship between
thought of speeding and feeling unsafe downtown.
Question 7b
Test for relationship between police contact and age group
Hypothesis
H0: There is no relationship between police contact and age group.
Versus
H1: There is a relationship between police contact and age group.
The study will employ chi-square since we are testing independence between two variables. .
The alpha value is 0.05
Table of results
Chi-Square Tests
Value df Asymp. Sig. (2-
sided)
Exact Sig. (2-
sided)
Exact Sig. (1-
sided)
Pearson Chi-Square 1.773a 1 .183
Continuity Correctionb 1.091 1 .296
Likelihood Ratio 1.778 1 .182
9 | P a g e

Statistics
Fisher's Exact Test .253 .148
Linear-by-Linear Association 1.738 1 .187
N of Valid Cases 50
a. 0 cells (0.0%) have expected count less than 5. The minimum expected count is 9.68.
b. Computed only for a 2x2 table
Table 10
The table above shows the results of the chi-square test. We can observe that the calculated p-
value is 0.25. This is greater than the alpha value which is 0.05. The decision therefore is that the
null hypothesis is accepted. The conclusion therefore is that there is a relationship between
police contact and age group.
Question 7b
Test for relationship between unsafe down town and age group.
Hypothesis
H0: There is no relationship between feeling unsafe down town and age group.
Versus
H1: There is a relationship between feeling unsafe down town and age group.
The study will employ chi-square since we are testing independence between two variables. .
The alpha value is 0.05
Table of results
Chi-Square Tests
Value df Asymp. Sig. (2-
sided)
Exact Sig. (2-
sided)
Exact Sig. (1-
sided)
Pearson Chi-Square .487a 1 .485
Continuity Correctionb .166 1 .684
Likelihood Ratio .486 1 .486
Fisher's Exact Test .567 .342
Linear-by-Linear Association .477 1 .490
10 | P a g e

Statistics
N of Valid Cases 50
a. 0 cells (0.0%) have expected count less than 5. The minimum expected count is 8.80.
b. Computed only for a 2x2 table
Table 11
The table above shows the results of the chi-square test. We can observe that the calculated p-
value is 0.57. This is greater than the alpha value which is 0.05. The decision therefore is that the
null hypothesis is accepted. The conclusion therefore is that there is a relationship between
feeling unsafe downtown and age group.
11 | P a g e