To qualify for poverty funds, legislators in a particular district in Atlanta had to show average household income for a family of four was $11,809.  A study was commissioned yielding the following: 

                                        n   =  53 households in district

                                        
 
x
¯
    =  $12,053 annual household income

                                        s   =  $4,320

 

At 
α
=
0
.
05
, test the claim that this sample was taken from a population of $11,809.

The alternative hypothesis (H1) is which of the following?
Question 1 options:

μ
⩽
11
,
809

μ
>
11
,
809

μ
<
11
,
809

μ
≠
11
,
809

μ
=
11
,
809


To solve this, we first need to understand the hypotheses for this hypothesis test.

Given:
- We are testing if the sample was taken from a population where the true mean household income is **$11,809**.
- The null hypothesis ($H_0$) will claim that the population mean is $11,809.
- The alternative hypothesis ($H_1$) reflects what we are trying to prove (whether the population mean differs from $11,809).

The null and alternative hypotheses are generally set up as follows:
- $H_0: \mu = 11,809$ (The population mean household income is $11,809).
- $H_1$: The alternative hypothesis is trying to test if the population mean **differs** from $11,809.

In this case, because the problem does not mention "greater than" or "less than," it seems like we are performing a **two-tailed test**. This means we are simply testing whether the mean is **different** from $11,809, either higher or lower.

Thus, the alternative hypothesis should be:
- **$H_1: \mu \neq 11,809$**.

**Answer**: The alternative hypothesis ($H_1$) is:
\[
\mu \neq 11,809
\]

Do you want further details about how to test this hypothesis or more clarification on two-tailed vs one-tailed tests?

Here are 5 related questions to expand the discussion:
1. How do you calculate the test statistic for this hypothesis test?
2. What does it mean to perform a two-tailed test?
3. How do you interpret the result of a hypothesis test if the null hypothesis is rejected?
4. What are Type I and Type II errors in hypothesis testing?
5. What is the p-value, and how is it used in hypothesis testing?

**Tip**: Always define your null and alternative hypotheses clearly before conducting any hypothesis test. This helps ensure you're testing the right claim.

To qualify for poverty funds, legislators in a particular district in Atlanta had to show average household income for a family of four was $11,809. A study was commissioned yielding the following: n = 53 households in district, x̄ = $12,053 annual household income, s = $4,320. At α=0.05, test the claim that this sample was taken from a population of $11,809. The alternative hypothesis (H1) is which of the following? Options: μ⩽11,809; μ>11,809; μ<11,809; μ≠11,809; μ=11,809.

This problem tests whether the average household income of $12,053 in a district differs from the poverty threshold of $11,809. Using a two-tailed hypothesis test with a significance level of α = 0.05, students will learn how to apply the z-test formula and interpret the results. Suitable for college-level statistics students.

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