Question

a personnel director in a particular state claims that the mean annual income is the same in one of the counties (county a) as it is in another county (county b). in county a, a random sample of 18 residents has an annual income of $41,100 and a standard deviation of $8400. in county b, a random sample of 8 residents has an annual income of $38,400 and a standard deviation of $5700. at α=0.10, answer parts (a) through (the problem mentions population variances are not equal. assume the samples are random and independent, and the populations are normally distributed.
(a) identify the claim and state ( h_0 ) and ( h_a ).
which is the correct claim below?
a. \the mean annual incomes in counties a and b are not equal.\
b. \the mean annual income in county a is less than in county b.\
c. \the mean annual incomes in counties a and b are equal.\
d. \the mean annual income in county a is greater than in county b.\
what are ( h_0 ) and ( h_a )?
the null hypothesis, ( h_0 ), is ( mu_1 = mu_2 ). the alternative hypothesis, ( h_a ), is ( mu_1
eq mu_2 ).
which hypothesis is the claim?
the alternative hypothesis, ( h_a )
the null hypothesis, ( h_0 )

Explanation:

1. For the correct claim: The personnel director claims the mean annual income is the same in County A and County B. Option C states this, so it's correct.
2. For \(H_0\) and \(H_a\): The null hypothesis (\(H_0\)) typically represents the claim of no difference (equality), so \(H_0: \mu_1 = \mu_2\). The alternative hypothesis (\(H_a\)) is the opposite, so \(H_a: \mu_1

eq \mu_2\) (since we're testing if they are not equal, as the claim is equality and we check against a difference).

1. For which hypothesis is the claim: The null hypothesis \(H_0\) states the mean incomes are equal, which matches the director's claim.

Answer:

1. Correct claim: C. "The mean annual incomes in Counties A and B are equal."
2. \(H_0\) and \(H_a\): \(H_0: \mu_1 = \mu_2\), \(H_a: \mu_1

eq \mu_2\)

1. Which hypothesis is the claim: The null hypothesis, \(H_0\)