This problem involves testing a claim about two population proportions at a significance level of $\alpha = 0.002$. Let's work through the solution step by step:

Step 1: Extract Information

• Null hypothesis ($H_0$): $p_1 = p_2$
• Alternative hypothesis ($H_a$): $p_1 < p_2$ (a left-tailed test)
• $n_1 = 667$, $p_1 = \frac{4\%}{100} = 0.04$
• $n_2 = 408$, $p_2 = \frac{5.4\%}{100} = 0.054$

The combined sample proportion ($p$) is calculated as: $p = \frac{x_1 + x_2}{n_1 + n_2}$ where $x_1 = n_1 \cdot p_1$ and $x_2 = n_2 \cdot p_2$.

Step 2: Calculate the Combined Proportion ($p$)

1. $x_1 = 667 \cdot 0.04 = 26.68 \approx 27$ (rounded to the nearest whole number)
2. $x_2 = 408 \cdot 0.054 = 22.032 \approx 22$
3. Total successes = $x_1 + x_2 = 27 + 22 = 49$
4. Total sample size = $n_1 + n_2 = 667 + 408 = 1075$
5. Combined proportion: $p = \frac{49}{1075} \approx 0.0456$

Step 3: Calculate the Standard Error (SE)

$SE = \sqrt{p \cdot (1 - p) \cdot \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}$

1. $1 - p = 1 - 0.0456 = 0.9544$
2. $\frac{1}{n_1} + \frac{1}{n_2} = \frac{1}{667} + \frac{1}{408} \approx 0.0015 + 0.00245 = 0.00395$
3. $SE = \sqrt{0.0456 \cdot 0.9544 \cdot 0.00395} \approx \sqrt{0.0001717} \approx 0.0131$

Step 4: Calculate the Test Statistic

The test statistic ($z$) is calculated as: $z = \frac{p_1 - p_2}{SE}$

$z = \frac{0.04 - 0.054}{0.0131} = \frac{-0.014}{0.0131} \approx -1.069$

Step 5: Calculate the P-Value

Since this is a left-tailed test, we find the probability corresponding to $z = -1.069$ using a standard normal table or software. The p-value is: $P(z < -1.069) \approx 0.142$

Step 6: Compare the P-Value to $\alpha$

• $\alpha = 0.002$
• $p\text{-value} = 0.142 > \alpha$

Thus, the p-value is greater than $\alpha$.

Step 7: Decision

We fail to reject the null hypothesis because the p-value is greater than the significance level.

Final Answers

1. Test statistic: $-1.069$ (rounded to 3 decimal places)
2. P-value: $0.1420$ (rounded to 4 decimal places)
3. Decision: Fail to reject the null hypothesis.

Would you like further clarification or details?

Related Questions:

1. What is the significance of using a pooled proportion in this hypothesis test?
2. How does the choice of significance level ($\alpha$) affect the conclusion of a hypothesis test?
3. How is a left-tailed test different from a right-tailed or two-tailed test in hypothesis testing?
4. Why do we use the standard error in the calculation of the test statistic?
5. How does the sample size influence the precision of the test statistic and the p-value?

Tip:

When calculating proportions and p-values, always ensure intermediate results are rounded only at the final step to maintain accuracy.