Questions: The National Institute of Mental Health published an article stating that in any one-year period, approximately 27 percent of American adults suffer from depression or a depressive illness. In a survey of 100 people in a certain town, the percentage of people suffering from depression or a depressive illness was actually different than 27 percent. The sample proportion was 28%. a. Choose the correct null and alternative hypothesis. Ho: p=0.27 and Ha: p ≠ 0.27 Ho: p ≥ 0.28 and Ha: p=0.28 Ho: p ≥ 0.28 and Ha: p<0.28 Ho: p=0.28 and Ha: p>0.28 Ho: p=0.27 and Ha: p<0.27 Ho: p=0.28 and Ha: p<0.28 Ho: p=0.27 and Ha: p>0.27

The National Institute of Mental Health published an article stating that in any one-year period, approximately 27 percent of American adults suffer from depression or a depressive illness.

In a survey of 100 people in a certain town, the percentage of people suffering from depression or a depressive illness was actually different than 27 percent. The sample proportion was 28%.

a. Choose the correct null and alternative hypothesis.
Ho: p=0.27 and Ha: p ≠ 0.27
Ho: p ≥ 0.28 and Ha: p=0.28
Ho: p ≥ 0.28 and Ha: p<0.28
Ho: p=0.28 and Ha: p>0.28
Ho: p=0.27 and Ha: p<0.27
Ho: p=0.28 and Ha: p<0.28
Ho: p=0.27 and Ha: p>0.27

Transcript text: The National Institute of Mental Health published an article stating that in any one-year period, approximately 27 percent of American adults suffer from depression or a depressive illness. In a survey of 100 people in a certain town, the percentage of people suffering from depression or a depressive illness was actually different than 27 percentage. The sample proportion was $28 \%$. a. Choose the correct null and alternative hypothesis. $H_{o}: p=0.27$ and $H_{a}: p \neq 0.27$ $H_{o}: p \geq 0.28$ and $H_{a}: p=0.28$ $H_{o}: p \geq 0.28$ and $H_{a}: p<0.28$ $H_{o}: p=0.28$ and $H_{a}: p>0.28$ $H_{o}: p=0.27$ and $H_{a}: p<0.27$ $H_{o}: p=0.28$ and $H_{a}: p<0.28$ $H_{o}: p=0.27$ and $H_{a}: p>0.27$

Solution

Solution Steps

Step 1: Hypotheses Formulation

We set up the null and alternative hypotheses as follows:

Null Hypothesis ($H_0$): $p = 0.27$
Alternative Hypothesis ($H_a$): $p \neq 0.27$

Step 2: Test Statistic Calculation

The test statistic $Z$ is calculated using the formula: \[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Substituting the values:

$\hat{p} = 0.28$
$p_0 = 0.27$
$n = 100$

We find: \[ Z = 0.2252 \]

Step 3: P-value Calculation

The P-value associated with the test statistic is calculated to be: \[ \text{P-value} = 0.8218 \]

Step 4: Critical Region Determination

For a two-tailed test at a significance level of $\alpha = 0.05$, the critical region is defined as: \[ Z < -1.96 \quad \text{or} \quad Z > 1.96 \]

Step 5: Conclusion

Since the calculated test statistic $Z = 0.2252$ does not fall into the critical region and the P-value $0.8218$ is greater than $\alpha = 0.05$, we fail to reject the null hypothesis.