To evaluate the manufacturer's claim that the defect rate is 5%, we conduct a hypothesis test using the sample data where 30 out of 500 widgets were defective (6% defect rate). The significance level is 5%.

• Null Hypothesis ((H_0)): The true defect rate (p = 0.05).
• Alternative Hypothesis ((H_a)): The true defect rate (p \neq 0.05) (two-tailed test).

Test Statistic (Z-test for proportion):

The test statistic is calculated as: [ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} ] where:

• (\hat{p} = \frac{30}{500} = 0.06) (sample proportion),
• (p_0 = 0.05) (claimed proportion),
• (n = 500) (sample size).

Substitute the values: [ z = \frac{0.06 - 0.05}{\sqrt{\frac{0.05 \times 0.95}{500}}} = \frac{0.01}{\sqrt{\frac{0.0475}{500}}} = \frac{0.01}{\sqrt{0.000095}} \approx \frac{0.01}{0.009747} \approx 1.026 ]

Critical Value and Decision:

• For a two-tailed test at (\alpha = 0.05), the critical z-values are (\pm 1.96).
• Since (|z| = 1.026 < 1.96), we fail to reject (H_0).

p-value Approach:

• The p-value for (z = 1.026) is (2 \times P(Z > 1.026) \approx 2 \times 0.152 = 0.304).
• Since (0.304 > 0.05), we fail to reject (H_0).

Conclusion:

There is insufficient evidence to reject the manufacturer's claim that the defect rate is 5%. The observed defect rate of 6% in the sample is not statistically significant at the 5% level. The difference between the sample rate (6%) and the claimed rate (5%) could reasonably occur due to random sampling variation.