Part 1 Of 8 Question 1 Of 1710 Points Two Teams Of Workers

Part 1 Of 8 Question 1 Of 1710 Pointsstwo Teams Of Workers Assemble A

Two teams of workers assemble automobile engines at a manufacturing plant in Michigan. A random sample of 145 assemblies from team 1 shows 15 unacceptable assemblies. A similar random sample of 125 assemblies from team 2 shows 8 unacceptable assemblies. Is there sufficient evidence to conclude, at the 10% significance level, that the two teams differ with respect to their proportions of unacceptable assemblies?

A. No, since the test value exceeds the critical value

B. No, since the test value does not exceed the critical value

C. No, since the p-value is less than 0.10

D. Yes, since the p-value is greater than 0.10

Paper For Above instruction

This paper examines whether two teams of workers differ significantly in their proportions of unacceptable automobile assemblies, based on a sample from each team. The problem involves testing the equality of two proportions at a significance level of 10%. The data indicates that from team 1, 15 unacceptable assemblies were found out of 145, while team 2 had 8 unacceptable assemblies out of 125.

The initial step involves setting up hypotheses: the null hypothesis (H₀) posits that the proportions of unacceptable assemblies are equal for both teams (p₁ = p₂). The alternative hypothesis (H₁) suggests that there is a difference between the two proportions (p₁ ≠ p₂). The test used is a two-proportion z-test, which involves calculating a combined proportion, the standard error, and then the z-statistic.

Calculations start with the sample proportions: p̂₁ = 15/145 ≈ 0.1034 and p̂₂ = 8/125 ≈ 0.064. The pooled proportion (p̂) is (15 + 8) / (145 + 125) = 23 / 270 ≈ 0.0852. The standard error is then calculated based on this pooled proportion and the sample sizes. Using the formula for the z-test statistic:

z = (p̂₁ - p̂₂) / √[p̂(1 - p̂)(1/n₁ + 1/n₂)]

Plugging in the values yields a test statistic of approximately 1.34. Critical value for a two-tailed test at α = 0.10 is approximately ±1.645. Since the test statistic (1.34) does not exceed the critical value (1.645), we fail to reject the null hypothesis.

The p-value corresponding to z = 1.34 is approximately 0.180, which exceeds the significance level of 0.10. Therefore, based on the data and the test conducted, there is insufficient evidence to conclude that the proportions of unacceptable assemblies differ between the two teams.

Thus, the correct conclusion aligns with option B: the test value does not exceed the critical value, and we fail to reject H₀ at the 10% significance level.

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