Question

A researcher is interested whether number of arrests vary across cities. Two random samples were collected. In city 1, the researcher sampled 185 individuals who had an average of 7.3 arrests with a standard deviation of 2.3 arrests. In city 2, the researcher sampled 160 individuals who had an average of 8.2 arrests with a standard deviation of 2.4 arrests. Test the null hypothesis at the .01 level of significance that the number of arrests does not vary across cities. In so doing, identify: (1) the research and null hypothesis, (2) the critical value needed to reject the null, (3) the decision that you made upon analyzing the data, and (4) the conclusion you have drawn based on the decision you have made.

Answer 1

for city 1

sample size=n1=185 sample mean=m1=7.3 sample SD=S1=2.3

for city 2

sample size =n2=160 sample mean =m2=8.2 sample SD=S2=2.4

since sample size for both sample is large so we will use Z test

1)

since we are testing that number of arrest in both cities are same or different so

2)

since test is two tailed and we are using Z test with level of significance 0.01

so we will have to critical values

such that

P(Z< critical value 1) =0.005 P(Z>critical value 2) =0.005

from Z table

P(Z<-2.58) =0.05 and P(Z>2.58) =0.005

so critical value 1=-2.58 and critical value 2=2.58

we reject H0 if test statistics <-2.58 or test statistics >2.58

3)

now test statistics is given by

since test statistics (-3.54) is less than critical value -2.58

Hence we reject H0

4)

there is enough statistical evidence to conclude that the number of arrests does vary across cities.