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A personnel director in a particular state claims that the mean annual income is the same in one of the​ state's counties​ (County A) as it is in another county​ (County B). In County​ A, a random sample of 17 residents has a mean annual income of $42,100 and a standard deviation of $8300.In County​ B, a random sample of 14 residents has a mean annual income of$37,900 and a standard deviation of $5700. At α=0.05,answer parts​ (a) through​ (e). Assume the population variances are not equal. Assume the samples are random and​ independent, and the populations are normally distributed.​(a) Identify the claim and stateUpper H 0H0andUpper H Subscript aHa. Which is the correct claim​ below? A.​"The mean annual incomes in Counties A and B are not​ equal." B.​"The mean annual incomes in Counties A and B are​ equal."This is the correct answer. C.​"The mean annual income in County A is greater than in County​ B." D.​"The mean annual income in County A is less than in County​ B."What are Upper H 0H0and Upper H Subscript aHa​? The null​ hypothesis,Upper H 0H0​,ismu 1 equals mu 2 .μ1=μ2. The alternative​ hypothesis,Upper H Subscript aHa​,ismu 1 not equals mu 2 .μ1≠μ2. Which hypothesis is the​ claim?  The alternative​ hypothesis, Upper H Subscript aHa The null​ hypothesis, Upper H 0H0Your answer is correct.​(b) Find the critical​ value(s) and identify the rejection​ region(s). Enter the critical​ value(s) below.

Question
A personnel director in a particular state claims that the mean annual income is the same in one of the​ state's counties​ (County A) as it is in another county​ (County B). In County​ A, a random sample of 17 residents has a mean annual income of $42,100 and a standard deviation of $8300.
In County​ B, a random sample of 14 residents has a mean annual income of
$37,900 and a standard deviation of $5700. At α=0.05,
answer parts​ (a) through​ (e). Assume the population variances are not equal. Assume the samples are random and​ independent, and the populations are normally distributed.
​(a) Identify the claim and state
Upper H 0H0
and
Upper H Subscript aHa.
 
Which is the correct claim​ below?
 
A.
​"The mean annual incomes in Counties A and B are not​ equal."
 
B.
​"The mean annual incomes in Counties A and B are​ equal."
This is the correct answer.
 
C.
​"The mean annual income in County A is greater than in County​ B."
 
D.
​"The mean annual income in County A is less than in County​ B."
What are Upper H 0H0
and Upper H Subscript aHa​?
 
The null​ hypothesis,
Upper H 0H0​,
is
mu 1 equals mu 2 .μ1=μ2.
 
The alternative​ hypothesis,
Upper H Subscript aHa​,
is
mu 1 not equals mu 2 .μ1≠μ2.
 
Which hypothesis is the​ claim?
 
 
The alternative​ hypothesis, Upper H Subscript aHa
 
The null​ hypothesis, Upper H 0H0
Your answer is correct.
​(b) Find the critical​ value(s) and identify the rejection​ region(s).
 
Enter the critical​ value(s) below.
check_circleAnswer
Step 1

According to the given question, in order to test that the annual income is the same in one of the​ state's counties​ (County A) as it is in another county​ (County B).

In County​ A, a random sample of 17 residents has a mean annual income of $42,100 and a standard deviation of $8300.
In County​ B, a random sample of 14 residents has a mean annual income of

$37,900 and a standard deviation of $5700.

Where we represent the entire data as:

Country A
Country B
mean
,3 7900
42100
mean
2
Standard deviation
Standard deviation
8300
s,= 5700
S
=
Sample size
Sample size
1 7
14
1
2
help_outline

Image Transcriptionclose

Country A Country B mean ,3 7900 42100 mean 2 Standard deviation Standard deviation 8300 s,= 5700 S = Sample size Sample size 1 7 14 1 2

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Step 2

We define the test hypothesis and the test statistics are as followsw;

 

Now we define the claim as:
The null hypothesis (there has no such difference);
Но : иА
A
Against the altemative hypothesis
as:
н
Now we define the test statistics when the population variances are not egual Assume the
samples are random andindependent, and the populations are normally distributed as
follows:
(x )
2
S
2
+
n
42100-37900
83002 5700
17
14
4200
2524.4935
1.6637
Therefore calculated t statistics is:
t 1.6637
help_outline

Image Transcriptionclose

Now we define the claim as: The null hypothesis (there has no such difference); Но : иА A Against the altemative hypothesis as: н Now we define the test statistics when the population variances are not egual Assume the samples are random andindependent, and the populations are normally distributed as follows: (x ) 2 S 2 + n 42100-37900 83002 5700 17 14 4200 2524.4935 1.6637 Therefore calculated t statistics is: t 1.6637

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Step 3

Where we can determine the d...

df
1
1
1
n-1
n2-1n
n
83002 57002
17
14
(5700
14-114
1 8300
17-1 17
1
=28
+
+
+
help_outline

Image Transcriptionclose

df 1 1 1 n-1 n2-1n n 83002 57002 17 14 (5700 14-114 1 8300 17-1 17 1 =28 + + +

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Hypothesis Testing

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