Question

A random sample of 17 police officers in Oak Park has a mean annual income of $35,800 and a standard deviation of $7,800. In Homewood, a random sample of 18 police officers has a mean annual income of $35,100 and a standard deviation of $7,375. Test the claim at α = 0.01 that the mean annual incomes in the two cities are not the same. Assume the population variances are equal.

a. Write down the type of test you will conduct.

b. Write down the null and alternative hypotheses.

c. Construct the test statistic.

d. Conduct the test.

e. What do you conclude?

Answer 1

Solution:

Given:

Police officers in Oak Park:

n₁ = 17

s₁ = 7800

Police officers in Homewood:

n₂ = 18

s₂ = 7375

Claim: the mean annual incomes in the two cities are not the same.

Level of significance = α = 0.01

the population variances are equal.

Part a. Write down the type of test you will conduct.

Since sample sizes are small ( < 30) , population standard deviations are unknown but assumed to be equal,

we use two sample independent t test for difference in two population means.

Part b.  Write down the null and alternative hypotheses.

Since claim is non-directional, this is two tailed test.

Vs

Part c.  Construct the test statistic.

where

Part d. Conduct the test.

thus

thus

Find t critical values:

df = n1 + n2 - 2 = 17 + 18-2 = 33

two tail area = 0.01

Since df = 33 is not given in t table, we look for its previous df = 30

thus t critical value = 2.750

Since this is two tailed test, we have two critical values: ( -2.750 , 2.750)

Decision Rule:

Reject null hypothesis H0, if absolute t test statistic value > t critical value = 2.750 , otherwise we fail to reject H0

Since t test statistic value = < t critical value = 2.750 , we fail to reject H0.

Part e. What do you conclude?

At 0.01 level of significance, we do not have sufficient evidence to support the claim that:  the mean annual incomes in the two cities are not the same.