To determine if the smoking rate in New York City has decreased, we will perform a hypothesis test for a population proportion. Here’s how the test can be structured:

Hypotheses:

• Null Hypothesis ($H_0$): $p = 0.12$ (the smoking rate is still 12%)
• Alternative Hypothesis ($H_a$): $p < 0.12$ (the smoking rate has decreased)

Test Type:

This is a one-tailed z-test for a population proportion.

Step 1: Sample Statistics

• Sample size ($n$): 70
• Number of successes ($x$): 9
• Sample proportion ($\hat{p}$): $\hat{p} = \frac{x}{n} = \frac{9}{70} \approx 0.1286$

Step 2: Test Statistic

The formula for the z-test statistic is: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ Where:

• $\hat{p}$ = sample proportion
• $p_0$ = hypothesized proportion (12% or 0.12)
• $n$ = sample size

Substitute the values: $z = \frac{0.1286 - 0.12}{\sqrt{\frac{0.12 \times (1 - 0.12)}{70}}}$

First, calculate the denominator: $\sqrt{\frac{0.12 \times 0.88}{70}} = \sqrt{\frac{0.1056}{70}} \approx \sqrt{0.00150857} \approx 0.03885$

Now calculate the z-value: $z = \frac{0.1286 - 0.12}{0.03885} \approx \frac{0.0086}{0.03885} \approx 0.2213$

Step 3: Determine the Critical Value

For a one-tailed test with a significance level ($\alpha$) of 0.05, the critical z-value is approximately $-1.645$.

Step 4: Decision Rule

• If $z \geq -1.645$, fail to reject the null hypothesis.
• If $z < -1.645$, reject the null hypothesis.

Step 5: Conclusion

The calculated z-value is $0.2213$, which is greater than $-1.645$. Therefore, we fail to reject the null hypothesis.

Final Interpretation:

There is not enough evidence to suggest that the smoking rate in New York City has decreased from 12%.

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Do you want further clarification or calculations? Here are 5 related questions to deepen understanding:

1. What would the test statistic be if the sample size was larger?
2. How does the significance level ($\alpha$) impact the hypothesis test?
3. What are Type I and Type II errors in this context?
4. How would a two-tailed test differ from the one-tailed test used here?
5. How can confidence intervals be used to support or refute these findings?

Tip: Always check the assumptions of a hypothesis test, such as the sample being random and large enough for normal approximation!