Homework #13 ME 368 - Engineering Measurements and Instrumentation Due in .pdf format, submitted in ecow2, by noon Thursday, Dec 10, 2009 Fall Semester 2009
HW Problem 13-1: (Objective: Combined chi-squared and student t analysis)
A sample of 5 ball bearings are pulled from a production line and their diameters are measured. The standard deviation of the diameters is 0.10 mm.
a) Determine the range that contains the population standard deviation in [mm] at 95% confidence.
2 2 [x/10] Using eq. 8.45 with ν = N – 1 = 4, Sx = 0.10 mm, and from table 8.8, / 2 = 11.1, 1 / 2 = 0.484, obtain 0.060 mm < σ < 0.29 mm (P = 95%).
b) What is the percent chance that σ is greater than 0.29 mm?
[x/10] Since the probability that σ is < 0.060 mm should be equal to the probability that σ is > 0.29 mm and the sum of these two probabilities sum to 5%, the percent chance that σ is greater than 0.29 mm is 2.5%.
c) How many bearings N should be measured so that the population mean diameter can be estimated to within 0.001 mm at 95% confidence? Use Sx = 0.29 mm in your calculation.
t S t 2 ,P95% x ,P95% t ,P95% [x/10] = 0.001 mm. Sx = 0.29 mm. thus =0.00345 N = . N N 0.0000119
Using the same reasoning as in quiz 10, choose t ,P95% =1.96 from table 8.4. Compute N = 1.9602 = 322 824, or to 2 significant digits, N = 3.2 x 105. 0.0000119
d) Given the combined probabilities involved in your answer to (c) {there is a probability associated with Sx as well as a probability associated with the student t value you used in (c)}, do you think the N value you computed in (c) will result in i. exactly 95% confidence in an estimate within 0.001 mm, as was the target. ii. better than 95% confidence in an estimate within 0.001 mm OR iii. worse than 95% confidence in an estimate within 0.001 mm and why?
[x/10] I think better than 95% confidence. It would be exactly 95% confidence if Sx was really 5 0.29 mm for the huge N = 3.2 x 10 sample. Actually, there is only a 2.5% chance that Sx is 0.29 mm or bigger, as determined in (b); there is a 97.5 % chance that Sx is smaller than 0.29 mm and 5 in this (likely) case, N = 3.2 x 10 results in better than 95% confidence. HW Problem 13-2: (Objective: Use student’s t distribution to determine whether 2 samples are from the same population)
Some students you may know have been measuring the speeds of suction cup darts fired from toy guns. Imagine 10 dart speeds were measured for each of 2 guns, with the resulting data given at http://homepages.cae.wisc.edu/~ssanders/me_368/homework_problem_statements/fall_2009/dart_speeds.xls
In this problem we will explore the following question: does one of these guns shoot faster than the other? Follow example 8.8 Dunn, and determine:
a) Statistically, which gun might be faster, gun 1 or gun 2, and why?
[x/10] By averaging each gun speed, we see that gun 2 is on average 0.87 m/s faster than gun 2.
b) Statistically, what is the percent confidence that the gun you identified in (a) is actually faster?
I suggest working the entire problem in Excel and then reporting your answer in such a way that the numbers and formulas you used are visible to the grader in the .pdf you turn in.
[x/50] Based on the data, we can be 92% confident that gun 2 is actually faster than gun 1. In other words, there is an 8% chance that if we took more data, we could find that the guns are the same speed or that gun 1 is actually faster. An Excel version of the solution is available at http://homepages.cae.wisc.edu/~ssanders/me_368/homework_solutions_with_problem_statemen ts/fall_2009/HW13_darts.xls
See first the ASSIGNED VERSION at right, answer in cell K23
It is instructive to play with the RANDOM VERSION at left and the N = 100 version below. It would be nice if someone made a LabVIEW version of this so N can be varied as well…