Suppose 20 Independent Observations Are Sampled from a Standard Normal Distribution

Homework 1 ( Due date: Sep 3rd. Wednesday)

Suppose 20 independent observations are sampled from a standard normal distribution. 1) What is the probability that the sample mean is between –0.5 and 0.2?

2) What is the probability that the sample standard deviation is greater than 1.2?

Homework 2 (Due date: Sep 17. Wednesday)

1) Assume that the simple statistical model Y= X + e

where Y observation, X measurand and e random normal error. One has measurements y for several parts (each part has one measurement) that produce a sample standard deviation s_y and several measurements on a single part that produce a sample standard deviation s. We also assume that all parts are from a stable process and the process variation and measurement variation are at the same level (all equal 1). Given sample sizes 10 for both samples, what is the probability that the estimate of process variation is 0? If the sample sizes increase to 30, what is the probability that the estimate of process variation is 0? If we pick an acceptable probability level 0.001, find the smallest sample sizes to have positive estimate of process variation.

2) In a typical industrial gage R&R study, we have following information: I=4, J=3, m=2. SSB=0.00080833, SSAB= 0.000725, SSE=0.00035, MSB=4.042*10^-4, MSAB=1.208*10^-4, MSE=2.917*10^-5. Sigma_reproducibility=9.014*10^-3, with Degree of freedom =4.035, Sigma_R&R=0.011 with degree of freedom =7.452. find the 95% confidence intervals for sigma ( repeatability, reproducibility, R&R) and GCR given U=7 and L=4. HW 3(Due date: Oct 1. Wednesday)

1)Point estimates for 0/1 contexts. Suppose that I=3 parts are inspected by J=4 operators, m=10 times. In the table below are sample fractions of "non-conforming" calls made by the operators. Please calculate the point estimates for σ_repeatability, σ_reproducibility and σ_R&R

operator 1 operator 2 operator 3 operator 4 part 1 .1 .2 .2 .3 part 2 .2 .3 .2 .3 part 3 .4 .4 .5 .5

2) Find d_2(n) for n=2, n=3 and n=16.

3) State the purposes of control charting. What is suggested by out of control signals? At first glance, the term “control chart” perhaps suggests some type of graph depicting continuous regulatory efforts. Is this perception accurate? Why or why not? Suggest a better term than “control chart”.

HW 4 (Due date: Oct 15. Wednesday) problems from text book 3.18 (a),(b) need to show me all steps to find control limits for xbar chart and s chart. 3.27 3.31

Computing tasks: 1) download and install qcc package in R 2) Page 72 table 3.2, please find k if n=10

HW 5 (Due date: Oct 29. Wednesday) qcc package

Please read the description of two data sets: pistonrings and orangejuice.

1) Create a retrospective xbar chart for piston rings data using first 25 samlpes . Suppose the first 25 samples are from a stable process and check the additional 15 samples(create a new xbar chart by adding 15 samples), what is your conclusion?

2) Create a retrospective p chart for orangejuice data and check the additional samples.(same as (1)) What is your conclusion?

3) Create a pareto chart and a fishbone diagram.

Examples:

1. Pareto Charts defect <- c(80, 27, 66, 94, 33) names(defect) <- c("price code", "schedule date", "supplier code", "contact num.", "part num.") pareto.chart(defect)

2. Fish bone diagram/ cause-and-effect diagram

cause.and.effect( cause=list(Measurements=c("Microscopes", "Inspectors"), Materials=c("Alloys", "Suppliers"), Personnel=c("Supervisors", "Operators"), Environment=c("Condensation", "Moisture"), Methods=c("Brake", "Engager", "Angle"), Machines=c("Speed", "Bits", "Sockets")), effect="Surface Flaws")

Problem 1: Suppose the products are normally distributed with mean 10 and standard deviation 1. Samples of size 5 are collected. Suppose that an all-ok ARL is about 370 and the quickest possible detection of a shift in mean is .5, what are control limits for a EWMA chart of xbar? For a CUSUM chart, what is the decision interval?

Problem2: Create an X/MR monitoring scheme for the following data set. We believe that the ideal “all-ok” ARL is 370 for our problem.

Sample 1 2 3 4 5 6 7 8

X 5 3 9 10 17 4 6 2 Problem 3: Find the quantile function and create a quantile plot for the data in problem 2. What is Q(0.37)? Also create a normal Q-Q plot and what is your conclusion?

Problem 4: From your textbook: 5.7

Qcc package

1) Create a ewma chart for pistonrings

2) Create a cusum chart for pistonrings

HW7(Due date: Nov 26. Wednesday)

Problem 1 Suppose we have a sample of n=30 observations with sample mean 5.7 and sample standard deviation 1.0. The specifications L and U are 2.0 and 9.8, respectively. Find 95% confidence intervals for process capability, C_p and C_pk. What is the assumption for your capability analysis?

Problem 2 Suppose we have a sample of n=30 observations with sample mean 5.7 and sample standard deviation 1.0. Find the 95% confidence interval for a single additional individual from the process. Find 95% tolerance interval for a fraction p=.9 of the distribution.

Problem 3: The constant is 2.145 for the tolerance interval in problem 2. Use a simulation study to confirm this number.

Problem 4: From your textbook: 5.18, 5.31

Problem 5: Analyze the following data

1) Construct a 95% confidence interval for the standard deviation of the noise.

2) Estimate main effects and fitted interactions.

3) Are these interactions statistically significant?( use 95% confidence level) 4) Is the difference in factor A main effects statistically significant? ( use 95% confidence level) factor A factor B observations

1 1 80,90,85

1 2 100,110,95

1 3 70,85,90

2 1 30,35,40

2 2 60,40,35

2 3 20,25,30