HW4p1_Statistics3_vanvlist

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University of Cincinnati, Main Campus *

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1120

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Industrial Engineering

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Apr 3, 2024

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pdf

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ENED 1120 – HW 4.1 – Spring 2024 Page 1 of 4 ENED 1120 – Spring 2024 HW 4.1: Central Limit Theorem and Confidence Intervals I NDIVIDUAL A SSIGNMENT : See the course syllabus for a definition of what constitutes an individual assignment. Task 1 (of 4) The population mean , μ, and standard deviation , σ, of dry time for a certain brand of latex paint have been 4.0 hrs. and .2 hrs. respectively. Fifty cans of latex paint were randomly selected, and a test was run to determine the amount of time it took the paint to dry. Answer the following questions, assuming the historical mean and standard deviation are still valid. Keep in mind, these problems are working with samples, so the axis you are plotting on is the mean of x. NOTE: Diagram as accurate as possible. You are welcome to do this on a separate piece of paper with the included area shaded and replace the image in Column 2 with your own image. Find Diagram Solution (show formulas or commands used and answer) (a) Probability that the mean dry time of the sample is less than 3.5 hrs. Z=3.5/0.2/sqrt50 Z=-17.67 0.0000000001 (b) Probability that the mean dry time of the sample is between 3.9 and 4.2 hrs. P(Z<-3.53) =0.0002 0.9999999-0.0002 =0.9997999
ENED 1120 – HW 4.1 – Spring 2024 Page 2 of 4 Task 2 (of 4) You work for a manufacturer of steel rods. Historically the process has resulted in rods with a normally distributed diameter with a mean , μ, of 3.0 cm and a standard deviation , σ, of .1 cm. Recently, one of your customers has complained that the rods are loose when used in their application. You pull a random sample of 10 rods and get the following diameters (in cm). [3.1, 3.0, 3.2, 3.0, 2.9, 2.8, 3.0, 3.2, 3.0, 2.8] a) What is the mean of the sampled data? The mean is 3 cm Based on the sampled data, and assuming the historical standard deviation has not changed, calculate the following: Find Diagram (remember to label axis) Solution (show formulas or commands used and answer) (b) Construct a two- sided 95% confidence interval for the population mean. Confidence Interval: (2.642, 3.358) (c) Construct a two- sided 90% confidence interval for the population mean. Confidence Interval: (2.709, 3.291) d) Which interval is wider and why does that make sense? The 90% confidence interval is wider, because the confidence interval is lower, which means it is less likely. e) Is the historical mean (μ = 3 cm) in the interval? It is in both of the intervals. f) What does this tell you about whether there has been a change in the rod diameter, and should the process be investigated further? There is a difference between the sample and historical mean, which may suggest that there was a change in rod diameter.
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