To determine what is incorrect in this solution, let's analyze each part systematically:

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a) Hypotheses:

• Null hypothesis ($H_0$): $P_1 - P_2 = 0$
  Correct
  The null hypothesis correctly states that there is no difference in the proportion of defective items between the two lots.

• Alternative hypothesis ($H_a$): $P_1 - P_2 \neq 0$
  Correct
  The alternative hypothesis correctly indicates a two-tailed test for whether there is a significant difference between the proportions.

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b) Test Statistic Calculation:

The test statistic formula for comparing two proportions is:

$z = \frac{(P_1 - P_2)}{\sqrt{\hat{P}(1 - \hat{P})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$

Where:

• $P_1 = \frac{10}{250} = 0.04$, $P_2 = \frac{18}{300} = 0.06$
• $\hat{P} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{10 + 18}{250 + 300} = 0.056$
• $n_1 = 250$, $n_2 = 300$

Now substituting:

$z = \frac{(0.04 - 0.06)}{\sqrt{0.056(1 - 0.056)\left(\frac{1}{250} + \frac{1}{300}\right)}} = \frac{-0.02}{\sqrt{0.056 \cdot 0.944 \cdot 0.00707}}$

$z = \frac{-0.02}{\sqrt{0.000374}} = \frac{-0.02}{0.0193} \approx -1.04$

The correct test statistic is approximately $-1.04$, not -1.06. This is a small numerical discrepancy.

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c) P-Value:

For a $z$-score of $-1.04$, the corresponding p-value for a two-tailed test can be found using a standard normal table or software.

• From the table, the area to the left of $z = -1.04$ is approximately $0.1492$.
• The total p-value for a two-tailed test is $2 \times 0.1492 = 0.2984$.

The correct p-value is approximately 0.2984, not 0.2892.

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d) Decision:

At a significance level of $\alpha = 0.02$:

• The p-value ($0.2984$) is much greater than $0.02$.
• Hence, we fail to reject the null hypothesis.

The conclusion is:

• "No, there is not sufficient evidence to support the claim that the quality of the two lots is different."

The answer "A. No" is correct.

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Corrections:

1. The test statistic should be $-1.04$ (not $-1.06$).
2. The p-value should be 0.2984 (not $0.2892$).

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Revised Answers:

a) $H_0: P_1 - P_2 = 0$, $H_a: P_1 - P_2 \neq 0$ (Correct)
b) Test statistic: $-1.04$ (Incorrect earlier, corrected here)
c) p-value: $0.2984$ (Incorrect earlier, corrected here)
d) Conclusion: $A$ (Correct)

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Do you want me to provide step-by-step explanations for the calculations?

Here are five relative questions to expand the understanding:

1. What is the formula for pooling proportions in a hypothesis test?
2. How does the choice of significance level ($\alpha$) affect the conclusion?
3. What are the implications of using a two-tailed vs. one-tailed test in this context?
4. How would the conclusion change if the significance level were set to $0.05$?
5. Can this hypothesis test be conducted without pooling the proportions? If yes, how?

Tip: Always verify calculations using a normal table or software for accurate p-values, as small numerical errors can impact conclusions.