Question

A personnel director in a particular state claims that the mean annual income is greater in one of the​ state's counties​ (county A) than it is in another county​ (county B). In County​ A, a random sample of 12 residents has a mean annual income of $ 42 comma 400 and a standard deviation of $ 8200. In County​ B, a random sample of 10 residents has a mean annual income of $ 37,900 and a standard deviation of $ 5300. At alpha=0.005​, answer parts​ (a) through​ (e). Assume the population variances are not equal. If​ convenient, use technology to solve the problem. '

​(a) Identify the claim and state Upper H 0 and Upper H Subscript a.

Which is the correct claim​ below?

A. ​"The mean annual income in county A is less than in county​ B." B. ​"The mean annual incomes in counties A and B are not​ equal." C. ​"The mean annual income in county A is greater than in county​ B." D. ​"The mean annual incomes in counties A and B are​ equal." What are Upper H 0 and Upper H Subscript a​? The null​ hypothesis, Upper H 0​, is mu 1 equals mu 2 . The alternative​ hypothesis, Upper H Subscript a​, is mu 1 not equals mu 2. Which hypothesis is the​ claim? The null​ hypothesis, Upper H 0 The alternative​ hypothesis, Upper H Subscript a ​(b) Find the critical​ value(s) and identify the rejection​ region(s). Enter the critical​ value(s) below. ​(Type an integer or decimal rounded to three decimal places as needed. Use a comma to separate answers as​ needed.) Select the correct rejection​ region(s) below. A. t less than minus t 0 B. t greater than t 0 C. negative t 0 less than t less than t 0 D. t less than minus t 0,  t greater than t 0 ​(c) Find the standardized test statistic. t= ​(Type an integer or decimal rounded to three decimal places as​ needed.) ​(d) Decide whether to reject or fail to reject the null hypothesis.

​(e) Interpret the decision in the context of the original claim. At the 0.5 % significance​ level, ___________ there is not there is enough evidence to support the personnel​ director's claim.

Answer 1

We will use a one sided welch's t-test of two independent population means for unequal variances.

Since the initial claim is the mean annual income is greater in one of the​ state's counties​ (county A) than it is in another county​ (county B).

​(a) Identify the claim

Which is the correct claim​ below?

C. ​"The mean annual income in county A is greater than in county​ B.

: VS

Null hypo: mean annual income in county A is greater than in county​ B

Vs

Alternative hypo: mean annual income in county A is greater than in county​ B

(b) Find the critical​ value(s) and identify the rejection​ region(s).

The critical is region the red area.

Critical Value : = 2.861 From t-dist tables

df   

(c) Find the standardized test statistic. t= ​(Type an integer or decimal rounded to three decimal places as​ needed.)

Test Stat =   

Test Stat : 1.552

​(d) Decide whether to reject or fail to reject the null hypothesis.

Since T.S. < C.V.

We do not reject the null hypothesis.

​(e) Interpret the decision in the context of the original claim.

At the 0.5 % significance​ level, there is not there is enough evidence to reject the personnel​ director's claim.