Conduct a full hypothesis test and determine if the given claim is supported or not supported at the 0.05 significance level. Round all amounts and standard scores to two decimal places.
-A manufacturer considers his production process to be out of control when defects exceed 3%. In a random sample of 85 items, the defect rate is 5.8% but the manager claims that this is only a sample fluctuation and production is not really out of control. Test whether the manufacturer's claim that production is out of control is supported or not supported.
A)supported
B)not supported

Question

Quizplus · Accepted Answer

Given: $n = 85$, sample proportion $p = 0.058$, hypothesized proportion $p_0 = 0.03$, $\alpha = 0.05$.1. Null hypothesis ($H_0$): The production process defect rate is 3% or less ($p_0 \leq 0.03$).2. Alternative hypothesis ($H_1$): The production process defect rate is more than 3% ($p_0 > 0.03$).Test statistic:$$ z = \frac{p - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} $$$$ z = \frac{0.058 - 0.03}{\sqrt{\frac{0.03(1-0.03)}{85}}} \approx \frac{0.028}{\sqrt{\frac{0.03(0.97)}{85}}} \approx \frac{0.028}{0.018} \approx 1.56 $$Critical value for $\alpha = 0.05$ in a one-tailed test is approximately 1.645.Since the calculated z-value (1.56) is less than the critical z-value (1.645), we fail to reject the null hypothesis.Conclusion: The manufacturer's claim that the production is not out of control is not supported by the sample evidence at the 0.05 significance level.

Conduct a Full Hypothesis Test and Determine If the Given