A Mixture Of Pulverized Fuel Ash And Portland Cement
A Mixture Of Pulverized Fuel Ash And Portland Cement To Be Used For Gr
A mixture of pulverized fuel ash and Portland cement to be used for grouting should have a compressive strength of more than 1300 KN/m2. The mixture will not be used unless experimental evidence indicates conclusively that the strength specification has been met. Suppose compressive strength for specimens of this mixture is normally distributed with σ = 56. Let μ denote the true average compressive strength. Let X denote the sample average compressive strength for n = 12 randomly selected specimens. Consider the test procedure with test statistic X and rejection region x ≥ 1331.26. mean of the test statistic KN/m2 standard deviation of the test statistic KN/m2 Using the test procedure of part (b), what is the probability that the mixture will be judged unsatisfactory when in fact μ = 1350 (a type II error)? (Round your answer to four decimal places.) What is the probability distribution of the test statistic when μ = 1350? mean of the test statistic KN/m2 standard deviation of the test statistic KN/m2 Using the test procedure of part (b), what is the probability that the mixture will be judged unsatisfactory when in fact μ = 1350 (a type II error)? (Round your answer to four decimal places.) How would you change the test procedure of part (b) to obtain a test with significance level 0.05? (Round your answer to two decimal places.) Replace 1331.26 KN/m2 with KN/m2. What impact would this change have on the error probability of part (c)? (Round your answer to four decimal places.) The probability that the mixture will be judged unsatisfactory when in fact μ = 1350 will change to . Consider the standardized test statistic Z = (X − 1300)/(σ/ √ n). What are the values of Z corresponding to the rejection region of part (b)? (Round your answer to two decimal places.) For a new design of the braking system, the true average braking distance at 40 mph is known to be 120 ft. The new design will be adopted only if data indicates a reduction. (c) What is the significance level for the region of part (b)? (Round your answer to four decimal places.) How would you change the region to achieve α = 0.001? (Round your answer to one decimal place.) Replace the limit(s) in the region with . (d) What is the probability that the new design is not implemented when its true average braking distance is 115 ft and the current region is used? (Round your answer to four decimal places.) (e) Define Z = (X − 120)/(σ/ √ n ). What is the significance level for {z: z ≤ −2.39}? For {z: z ≤ −2.69}, (Round your answers to four decimal places.) For {z: z ≤ −2.39} and {z: z ≤ −2.69}: Let Z be a standard normal variable with the null hypothesis. What is the significance level? (Round your answers to four decimal places.) For Ha: μ > μ0, rejection region z ≥ 1.71 (b) Ha: μ ≠ μ0, df=18, rejection region t ≥ 3.610 (b) Ha: μ
Paper For Above instruction
Analysis of Statistical Testing in Engineering and Manufacturing Contexts
Statistical hypothesis testing is fundamental in engineering and manufacturing for ensuring product quality and compliance with set standards. This paper explores the application of hypothesis testing procedures in the context of materials strength assessment, brake system design, and quality control in manufacturing, illustrating how statistical tools inform critical decision-making processes.
1. Compressive Strength of Pulverized Fuel Ash and Portland Cement Mixture
In the context of grouting materials, ensuring that a mixture meets a minimum compressive strength threshold is crucial. The problem provides a scenario where the strength, measured across samples, is assumed to follow a normal distribution with a known standard deviation σ = 56 KN/m². The hypothesis test aims to determine whether the true mean strength μ exceeds 1300 KN/m², based on a sample of size n=12 with a sample mean X. The rejection region for this test is set at X ≥ 1331.26 KN/m², indicating a one-sided test focusing on whether the mixture's strength is sufficient.
The distribution of the test statistic under the null hypothesis is crucial for understanding error probabilities. When the true mean strength μ = 1350 KN/m², the test statistic Z follows a standard normal distribution, because of the known σ and the sample size. Calculating the probability of a Type II error (β), which is failing to reject the null hypothesis when μ is actually 1350 KN/m², involves determining the likelihood that the test statistic X does not exceed the critical value when the true mean is higher.
2. Adjusting Test Procedures for Desired Significance Levels
To control the significance level α at 0.05, the critical value must be adjusted. The new critical value corresponds to the 95th percentile of the normal distribution with the specified standard deviation and sample size. This adjustment affects the Type II error probability, potentially increasing it, which illustrates the trade-off between sensitivity and specificity in hypothesis testing.
3. Standardized Test Statistic and Its Interpretation
The standardized test statistic Z = (X − 1300)/(σ/√n) allows for evaluating the strength of evidence against the null hypothesis. Corresponding critical Z-values are derived from standard normal distribution tables, aligning with the set rejection regions. This standardization facilitates comparison across different studies and experimental settings.
4. Brake System Design and Reduction in Braking Distance
In evaluating new brake system designs, hypothesis testing determines whether the observed reduction in braking distance is statistically significant. By setting appropriate rejection regions, the manufacturer can decide whether to proceed with implementation. Adjustments to the rejection thresholds enable control over the significance level, such as α = 0.001, ensuring rigorous standards.
5. Normal Distribution Testing in Quality Control of Nickel Plates
Quality control involves testing the proportion of defective nickel-hydrogen batteries. Using a binomial model approximated by a normal distribution, the hypothesis test assesses whether the proportion of blistered plates exceeds a threshold, for example, 15%. The critical region and test statistic are calculated accordingly. The analysis extends to different sample sizes and error probabilities, informing production and testing strategies to maintain high quality standards.
Conclusion
The application of hypothesis testing in engineering and manufacturing contexts provides vital insights into product quality, safety, and performance. Properly selecting significance levels, understanding error probabilities, and adjusting testing procedures are essential for making informed decisions that balance risks and benefits. These statistical tools, when correctly applied, help ensure that engineering designs and manufacturing processes meet rigorous standards, ultimately safeguarding consumer safety and maintaining product integrity.
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