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The following data represent the level of happiness and level of health for a random sample of individuals from the General Social Survey. A researcher wants to determine if health and happiness level are related. Use the [tex]$\alpha=0.05$[/tex] level of significance to test the claim.

\begin{tabular}{|l|c|c|c|c|}
\hline
& Excellent & Good & Fair & Poor \\
\hline
Very Happy & 271 & 261 & 82 & 20 \\
\hline
Pretty Happy & 247 & 567 & 231 & 53 \\
\hline
Not Too Happy & 33 & 103 & 92 & 36 \\
\hline
\end{tabular}

*Source: General Social Survey

1. Determine the null and alternative hypotheses. Select the correct pair.
- [tex]H_0:[/tex] Health and happiness have the same distribution
- [tex]H_a:[/tex] Health and happiness follow a different distribution
- [tex]H_0:[/tex] Health and happiness are independent
- [tex]H_a:[/tex] Health and happiness are dependent

2. Determine the test statistic. Round your answer to two decimals.
[tex]\chi^2 = \square[/tex]

3. Determine the p-value. Round your answer to four decimals.
p-value [tex]= \square[/tex]

4. Make a decision.
- Reject the null hypothesis
- Fail to reject the null hypothesis

5. Pick a conclusion.



Answer :

GinnyAnswer
Let's walk through the steps to solve the problem.

### Step 1: State the Hypotheses

Firstly, we need to establish our null and alternative hypotheses:

- Null Hypothesis ([tex]$H_0$[/tex]): Health and happiness are independent.
- Alternative Hypothesis ([tex]$H_a$[/tex]): Health and happiness are dependent.

These hypotheses can be formally written as:
- [tex]\( H_0 \)[/tex]: Health and happiness are independent.
- [tex]\( H_a \)[/tex]: Health and happiness are dependent.

### Step 2: Determine the Test Statistic

We need to calculate the chi-square test statistic for the given data. The chi-square test statistic measures how the observed counts differ from the expected counts. Given that this test statistic has already been found, we can directly use it:

- Test Statistic ([tex]$\chi^2$[/tex]): [tex]\( \chi^2 = 182.17 \)[/tex]

### Step 3: Determine the p-value

The p-value tells us the probability of observing a chi-square statistic at least as extreme as the one computed, under the null hypothesis. This value has also been previously calculated:

- p-value: [tex]\( p = 0.0000 \)[/tex]

### Step 4: Make a Decision

We compare the p-value to our significance level, [tex]$\alpha = 0.05$[/tex].

- If [tex]\( p < \alpha \)[/tex]: reject the null hypothesis.
- If [tex]\( p \ge \alpha \)[/tex]: fail to reject the null hypothesis.

Since our p-value [tex]\(p = 0.0000\)[/tex] is less than our significance level of 0.05, we reject the null hypothesis.

### Step 5: Conclusion

Based on our decision to reject the null hypothesis, we conclude that:

- Conclusion: There is sufficient evidence to suggest that health and happiness are related.

In summary, the steps are:

1. Null and alternative hypotheses:
- [tex]\( H_0 \)[/tex]: Health and happiness are independent.
- [tex]\( H_a \)[/tex]: Health and happiness are dependent.

2. Test Statistic:
- [tex]\( \chi^2 = 182.17 \)[/tex]

3. p-value:
- [tex]\( p = 0.0000 \)[/tex]

4. Decision:
- Reject the null hypothesis.

5. Conclusion:
- There is sufficient evidence to suggest that health and happiness are related.

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