This document discusses three statistical tests: 1) A one-sample z-test is used to test if the proportion of breast cancer in female beauticians who use hair dye is greater than the national average. The null hypothesis is rejected, showing hair dye use is linked to higher breast cancer risk. 2) A two-sample z-test compares the proportion of children who died in car accidents when wearing a seatbelt vs not. The null hypothesis cannot be rejected, so seatbelt use may not impact children's risk of death in accidents. 3) A chi-squared test examines the relationship between bicycle helmet use and head injuries. The null hypothesis is rejected, demonstrating helmet use reduces the risk of head injury

1. Comparison of population proportion
One-sample problem
A group of investigators wish to explore the
relationship between the use of hair dyes and the
development of breast cancer in women. A sample
of n =1000 female beauticians 40–49 years of age
is identified and followed for five years. After five
years, x= 20 new cases of breast cancer have
occurred. It is known that breast cancer incidence
over this time period for national average in this
age group is π𝑜= 7=1000. We wish to test the
hypothesis that using hair dyes increases the risk
of breast cancer (a one-sided alternative)

2.  A one-sided test with 𝐻𝐴=π>
7
1000
 Using the conventional choice of α= 0.05 leads
to the rejection region z >1.65
 From the data P=
20
1000
=0.02
 leading to a ‘‘z score’’ of: Z=
0.02−0.007
0.007(0.993)/1000
=4.93
 (the observed proportion p is 4.93 standard
errors away from the hypothesized value of
π𝑜= 7=1000.
 Since the computed z score falls into the
rejection region 4.93 >1.65,the null hypothesis is
rejected at the 0.05 level chosen. In fact, the
differenceis very highly significant.

3. Comparison of two independent groups
Pooled proportion
In the sample of 123 children who were wearing a seat belt at the
time of the accident 3 died. Therefore 𝑃 1=
3
123
=0.024
In the sample of 290 children who were not wearing a seat belt
13 died. Consequently 𝑃 2=
13
290
=0.045

5. Conclusion :According to appendix A (Z Score
Table- chart value corresponds to area below z
score), The P-value of the test is
0.312.Therefore we cannot reject the null
hypothesis. The sample collected in this
particular study, do not provide evidence that
the proportion of children dying differ between
those who were wearing seat belts and those
who were not.

6. Chi-squared test

7. The results of the study investigating the
effectiveness of bicycle safety helmet in
preventing head injury consist of a random
sample 793 individuals who were involved in
the bicycle accident during a specific year
period. Of the 793 individuals who were
involved in bicycle accident,147 were wearing
safety helmets at the time of the incident and
646 were not. Among those wearing
helmets,17 suffered head injury requiring the
attention of doctors, whereas the remaining
130 did not; among the individuals not wearing
safety helmets ,218 sustained serious head
injuries, and 428 did not.
Question :Construct 2X2 table and test
hypothesis

8. Observed values
Expected values

10. For 𝑋2
distribution (appendix D) with 1 degree
of freedom, we re reject the null hypothesis
and conclude that the proportion of individuals
suffering head injuries are not identical in the
two proportions. Among persons involved in
bicycle accidents, the use of a safety helmet
reduces the incidence of head injury.