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.