Statistics Assignment: Hypothesis Testing and Analysis

This document presents a solved statistics assignment that focuses on hypothesis testing. The assignment includes three distinct problems. The first problem involves comparing mean juvenile crime rates between areas with high and low collective efficacy, utilizing a t-test to determine statistical significance. The second problem examines the impact of a program on police officer mental health, employing a z-test and p-value analysis to assess the program's effectiveness. The third problem investigates the relationship between prior records and sentence length, using correlation analysis to determine if a significant relationship exists. Each problem includes the formulation of null and alternative hypotheses, the application of appropriate statistical tests, the calculation of test statistics, and the interpretation of results to draw conclusions about the hypotheses.

Statistics
Statistics
Student name:
Tutor name:
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Statistics
QUESTION 1
Are areas with greater levels of collective efficacy having significantly different mean juvenile
crime rates?
HCE: High collective efficacy
LCE: Low collective efficacy
HCE LCE
mean 37.52 52.44
variance 8.2 14.25
n 100 100
Table 1
Hypothesis
H0: μ1 = μ2
Versus
H1: μ1 ≠ μ2
The significance level alpha = 0.05
Test statistic T = ( X 1−X2 )
√ s1
2
n1
− s2
2
n2
Test statistic T = ( 52.44−37.52 )
√ 14.25
100 − 8.2
100
Test statistic T = 14.92
0.245967 =60.66
However t0.95 , 198=1.96
It can be observed that T > t0.95 , 198 that is 60.66 > 1.96
The null hypothesis is rejected. The conclusion is that the areas with greater levels of collective
efficacy will have significantly different mean juvenile crime rate.
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Statistics
QUESTION 2
Hypothesis
H0: The program had no impact on police officer mental health
Versus
H1: The program had significant impact on police officer mental health.
( x−μ )=6.72
σ = 2.14
n = 75
The confidence level set is 95%
The Z-statistic is applied here because the sample size is greater than 30 and the standard
deviation is also known. The distribution of the population is therefore assumed to be normal.
formula; Z= x−μ
σ
√ n
Z= 6.72
2.14
√ 75
= 6.72
0.247 =27.21
Z=27.21
P –value corresponding to Z- value of 27.21 is 0.00001.
The results showed that the impact of the program had significant impact on police
officer mental health (p = 0.00). This is because the p-value computed was less than the level of
significance (0.05) thus making us to reject the null hypothesis and accept the alternative. The
conclusion is that the program had significant impact on police officer mental health.
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Statistics
QUESTION 3
Hypothesis
H0: There is no relationship between prior record and sentence length.
Versus
H1: There is a significant relationship between prior record and sentence length.
The confidence level set is 95%
Table of results
Correlations
Prior record Sentence
length
Prior record Pearson Correlation 1 .390
Sig. (2-tailed) .089
N 20 20
Sentence length Pearson Correlation .390 1
Sig. (2-tailed) .089
N 20 20
Table 2
The p-value (0.09) computed was greater than the level of significance (0.05) thus
making us to accept the null hypothesis and reject the alternative. It is concluded that there is no
relationship between prior record and sentence length.
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