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To solve this hypothesis test, let's go through each part of the problem step-by-step.

### Given Information
- The initial percentage of smokers was estimated to be about \( 30\% \).
- After new smoking cessation programs, you believe the percentage has **reduced**.
- A sample of 2246 people was surveyed, and 619 of them smoke.
- Significance level \( \alpha = 0.05 \).

### Problem Breakdown

#### (a) Identify the Null and Alternative Hypotheses
1. **Null Hypothesis ( \( H_0 \) ):** The percentage of smokers has not reduced, i.e., it is still \( 30\% \).
   \[ H_0: p = 0.30 \]
   
2. **Alternative Hypothesis ( \( H_1 \) ):** The percentage of smokers has reduced.
   \[ H_1: p < 0.30 \]

#### (b) Type of Hypothesis Test
Since the claim is that the percentage of smokers has **reduced**, we are testing for a decrease. This calls for a **left-tailed test**.

#### (c) Significance Level
The significance level \( \alpha \) is given as **0.05**.

#### (d) Calculate the Test Statistic
1. **Sample Proportion ( \( \hat{p} \) ):**
   \[
   \hat{p} = \frac{619}{2246} \approx 0.2756
   \]

2. **Standard Error ( \( SE \) ):**
   \[
   SE = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.30 \times (1 - 0.30)}{2246}} \approx 0.0096
   \]

3. **Z-Test Statistic ( \( Z \) ):**
   \[
   Z = \frac{\hat{p} - p_0}{SE} = \frac{0.2756 - 0.30}{0.0096} \approx -2.52
   \]

#### (e) Calculate the p-Value
For a left-tailed test, we need to find the p-value corresponding to \( Z = -2.52 \).
Using the Z-table, a Z-score of -2.52 gives a p-value of approximately **0.0059**.

#### (f) Do You Reject the Null Hypothesis?
Since the p-value (0.0059) is **less than** the significance level \( \alpha = 0.05 \), we **reject the null hypothesis**.

#### (g) Conclusion
Since we rejected the null hypothesis, we conclude:
> There is sufficient evidence to warrant rejection of the claim that the percentage of smokers is less than 30%.

---

### Summary of Answers
1. **(a)** \( H_0: p = 0.30 \); \( H_1: p < 0.30 \)
2. **(b)** Left-tailed
3. **(c)** Significance level: 0.05
4. **(d)** Test statistic: \( Z \approx -2.52 \)
5. **(e)** p-value: 0.0059
6. **(f)** Reject the null hypothesis.
7. **(g)** Conclusion: There is sufficient evidence to warrant rejection of the claim that the percentage of smokers is less than 30%.

Would you like more details on any specific part? Here are some further questions you might consider:

1. How is the standard error in hypothesis testing calculated?
2. Why do we use a left-tailed test in this scenario?
3. What does rejecting the null hypothesis mean in practical terms?
4. How do we interpret the p-value in the context of this test?
5. What would happen if the significance level were lowered to 0.01?

**Tip:** Remember that a p-value less than the significance level indicates strong evidence against the null hypothesis.

Past studies have indicated that the percentage of smokers was estimated to be about 30%. Given the new smoking cessation programs that have been implemented, you now believe that the percentage of smokers has reduced. You randomly surveyed 2246 people and found that 619 smoke. Use a 0.05 significance level to test the claim that the percentage of smokers has reduced.

This problem explores hypothesis testing for proportions, testing the claim that smoking rates have reduced below 30% based on survey data from 2246 people with a significance level of 0.05. Learn how to set up null and alternative hypotheses, conduct a left-tailed test, and calculate the Z-test statistic and p-value for statistical inference.

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