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 6 residents has a mean annual income of $41,100 and a standard deviation of $8300. In County​ B, a random sample of 8

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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 6 residents has a mean annual income of $41,100 and a standard deviation of $8300. In County​ B, a random sample of 8 residents has a mean annual income of $39,100 and a standard deviation of $5300. At α=0.10​, 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
H0
and
Ha.
 
Which is the correct claim​ below?
 
A.
​"The mean annual income in county A is less than in county​ B."
 
B.
​"The mean annual income in county A is greater than in county​ B."
 
C.
​"The mean annual incomes in counties A and B are not​ equal."
 
D.
​"The mean annual incomes in counties A and B are​ equal."
 
What are H0 and Ha?
 
Which hypothesis is the​ claim?
 
The alternative​ hypothesis, Ha
 
The null​ hypothesis,  H0
 
​(b) Find the critical​ value(s) and identify the rejection​ region(s).
 
(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<−t0, t>t0
 
B.
t<−t0
 
C.
−t0<t<t0
 
D.
t>t0
 
​(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
10%
significance​ level,
there is not/
there is
enough evidence to support the personnel​ director's claim.
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