Question

An economist wants to test if there is any difference in income between Escambia county and Miami-Dade county in Florida. To do this, she collects random samples of residents from both these counties. The information collected from these samples is tabulated below. Assume that both the populations are normally distributed with equal population variances.

Escambia County (Population 1) Miami-Dade County (Population 2)
n1 = 11 n2 = 15
Mean= 36,700 Mean = 34 , 700
Stdv = 7800 Stdv = 7375

Required:
a. Set up the null and alternative hypotheses to test the economist’s claim.
b. Find the value of the standardized test statistic.
c. At 5% significance level find the rejection region and state your decision

Answer 1

a) As this is a two-tailed test (significant difference), the null and alternative hypothesis are:

`H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2\\eq 0`

b) Test statistic t=0.66

c) Critical value tc=2.064.

The rejection region is defined by |t|>2.064.

As |t|=0.66<2.064, the null hypothesis failed to be rejected.

There is not enough evidence to support the claim that there is significant difference in income between Escambia county and Miami-Dade county.

Explanation:

This is a hypothesis test for the difference between populations means.

The claim is that there is significant difference in income between Escambia county and Miami-Dade county.

Then, the null and alternative hypothesis are:

`H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2\\eq 0`

The significance level is 0.05.

The sample 1 (Escambia county), of size n1=11 has a mean of 36700 and a standard deviation of 7800.

The sample 2 (Miami-Dade county), of size n2=15 has a mean of 34700 and a standard deviation of 7375.

The difference between sample means is Md=2000.

`M_d=M_1-M_2=36700-34700=2000`

The estimated standard error of the difference between means is computed using the formula:

`s_(M_d)=\sqrt{(\sigma_1^2)/(n_1)+(\sigma_2^2)/(n_2)}=\sqrt{(7800^2)/(11)+(7375^2)/(15)}\\\\\\s_(M_d)=√(5530909.091+3626041.667)=√(9156950.758)=3026.045`

Then, we can calculate the t-statistic as:

`t=(M_d-(\mu_1-\mu_2))/(s_(M_d))=(2000-0)/(3026.045)=(2000)/(3026.045)=0.66`

The degrees of freedom for this test are:

`df=n_1+n_2-1=11+15-2=24`

This test is a two-tailed test, with 24 degrees of freedom and t=0.66.

The critical value for this level of significance (0.05), a two-tailed test and 24 degrees of freedom is tc=2.064.

As the test statistic is below the critical value, it falls in the acceptance region.

The null hypothesis failed to be rejected.

There is not enough evidence to support the claim that there is significant difference in income between Escambia county and Miami-Dade county.