**Quit smoking:** In a survey of 

444

 HIV-positive smokers, 

205

 reported that they had used a nicotine patch to try to quit smoking. Can you conclude that less than half of HIV-positive smokers have used a nicotine patch? Use the 

=α0.05

 level of significance and the critical value method.

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Part 1 of 5

State the appropriate null and alternate hypotheses.

*H*0

: 



*H*1

: 

This hypothesis test is a ▼(Choose one) test.

Let's start by setting up the null and alternative hypotheses for this problem:

### Given Information:
- Sample size (\( n \)) = 444
- Number of people who used a nicotine patch = 205
- We want to determine if less than half of HIV-positive smokers have used a nicotine patch.
- Significance level (\( \alpha \)) = 0.05

### Step 1: State the Hypotheses
Let \( p \) be the proportion of HIV-positive smokers who have used a nicotine patch.

- **Null hypothesis (\( H_0 \))**: \( p \geq 0.5 \)  
  This states that at least half of HIV-positive smokers have used a nicotine patch.

- **Alternative hypothesis (\( H_1 \))**: \( p < 0.5 \)  
  This states that less than half of HIV-positive smokers have used a nicotine patch.

### Step 2: Determine the Test Type
This is a **left-tailed** test because we are testing if the proportion is less than 0.5.

Let me know if you want further details on the calculation steps or if you have any questions.

### Related Questions
1. What is the formula for calculating the test statistic for a proportion hypothesis test?
2. How do we find the critical value for a left-tailed test at a 0.05 significance level?
3. What role does the p-value play in hypothesis testing, and how is it related to the critical value method?
4. How do sample size and variability impact the power of a hypothesis test?
5. What are Type I and Type II errors in hypothesis testing, and how do they relate to this scenario?

### Tip
Always sketch the standard normal curve to visualize the rejection region and where your test statistic might fall.

In a survey of 444 HIV-positive smokers, 205 reported that they had used a nicotine patch to try to quit smoking. Can you conclude that less than half of HIV-positive smokers have used a nicotine patch? Use the α = 0.05 level of significance and the critical value method.

Perform a hypothesis test to determine if less than half of HIV-positive smokers have used a nicotine patch. Analyze a sample of 444 smokers, with 205 using the patch, using a left-tailed test at a significance level of α = 0.05. Understand key concepts like hypothesis testing, critical values, and proportions.

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