Statistics Exam 3

Statistical Process Control

Almost all processes have built in variability and are composed of subprocesses. SPC charts monitor different measures of variability over a period of time and compares performance variability to historical performance variability. Components of SPC chart: Upper Control Limit: +3std from CL Central Line: Avg. measured value LCL: -3std from the CL If a pattern violates a run rule the process is out of control. Run Rules: 1. Point is greater/less than 3 std 2. 4/5 points are greater/less than 2std on same side 3. 4/5 point are greater/less than 1 std on same side 4. 9 points on the same side of the CL 5. 6 points all increasing or decreasing. 6. 14 points all alternating up and down If there is a violation the process should be examined for the cause common causes are: worn tooling/machinery, not maintaining their settings or improperly set tooling or machinery, broken tooling/machinery, or the raw material is inconsistent substituted, or expired. Other causes may require additional analysis by using the Ishikawa charts

Ishikawa Charts

Also known as fishbone chart or cause/effect chart. Its a graphical tool used to assist in the identification of the cause of an unstructured problem Components: Backbone Mayor factors: on bones of backbone Minor factors: on boned of the mayor factor bones

If you have three data sets one with n1=15, n2=28, and n3=32 which test would you use to means of the data sets?

Analysis of variance (ANOVA) is an inferential method that is used to test the equality of 3 or more population means.

If any of three data sets are non-normal, the appropriate comparison of means hypotheses test would be?

Anova Test

Fitting a straight line to a given data by the method of least squares:

Calculate: ∑x ∑y ∑xy ∑x² Plug them into the normal equations: here 'n' is the number of observations. ∑y = an + b∑x ∑xy = a∑x + b∑x² Solve the system for 'a' and 'b' and plug it into the prediction equation: y = a + bx

Building x-bar and R-bar SPC charts

Calculating 's' may be to difficult to perform on the shop floor, so R is frequently substituted. Select a sample size and collect historical data for 20-25 periods x-bar and the range is calculated for each sample period x-bar is the sample avg. for each period R is the range of the sample for each period x-barbar is the avg. of the x-bar R-bar is the avg. of the Ranges X-bar Chart: CL = x-barbar UCL = x-barbar+(A2)(R-bar) LCL = x-barbar-(A2)(R-bar) ±1 sigma limit = CL±1/3(A2)(R-bar) ±2 sigma limit = CL±2/3(A2)(R-bar) where A2 is a constant from table 9 based on sample size R-Bar Chart: CL = R-bar UCL = (D4)R-bar LCL = (D3)R-bar not commonly used 1 sigma intermediate = CL+1/3(UCL-CL) 2 sigma intermediate = CL+2/3(UCL-CL) D3 and D4 are constants from table 9 based on sample size.

For problems 32 to 35, brand A company claims that its oil dispersion product for contaminated environments is more effective than brands B and C. Use the data listed below from parts per million contamination to determine whether or not there is evidence to support that the companies claims are in fact true at 0.05: Brand A-22, 25, 32, 18, 23, 15, 30, 27, 19, 23 Brand B-19, 22, 18, 29, 28, 32, 17, 33, 28, 20 Brand C-30, 29, 25, 24, 15, 27, 30, 27, 18, 32

Comparing populations so use an H test. What is the critical value at alpha=0.05? H test uses the x² table for DOF = # of sets - 1 Critical value = 5.991 What is the test statistic? Combine your data sets in ascending order identifying the rank of each value and which data set the value comes from. Then sum all the ranks for each data set and plug into our test statistic. Use the x² table to find your critical value as well as the rejection region approach. We get: 0.904 In terms of effectiveness in English, what does this mean? a. The company is making a false claim b. The company is making a true claim c. There is insufficient data to assess d. I need to take this course again ANS: A

The critical value at alpha=0.05 for an H test with four data set groups is.

DOF is the # of data sets - 1 so 4-1 = 3 so x² = 7.815

ANOVA test

Determines if one or more alternatives is different than the others. Based on a ratio of the variance between and within the different alternatives. If the variation between is large and variance within is small the ratio is large. If the variation between is small and the variance within is large the ratio is small. When the ratio is large then it is more likely there is a difference among alternatives. Hypothesis tests: H₀: Means are identical H_a: Means are not identical 1. Calculate the sum of squares total 2. Calculate the sum of squares between 3. Calculate the sum of squares within 4. Calculate the mean square between 5. Calculate the mean square within 6. Calculate the F statistic.

For 21 to 24, you have an x bar chart with a CL of 2.0, a UCL of 3.5 and a LCL of 0.5 and intermediate control limits and an R chart with a CL of 2.0, an UCL of 3.5 and a LCL of 0.5.

Draw the graph. For period 1, if you have the following sample determine whether or not you are in control 1.7, 2.2, 1.9, 1.2 a. X bar and R in control b. X bar in control, R out of control c. X bar out of control, R in control d. X bar and R out of control Ans: A X bar and R in control, because the samples lie between the LCL (0.5) and the UCL (3.5) For period 2, if you have the following sample determine whether or not you are in control 3.2, 3.5, 3.1, 3.3 a. X bar and R in control b. X bar in control, R out of control c. X bar out of control, R in control d. X bar and R out of control ANS: Ans: A X bar and R in control, because the samples lie between the LCL (0.5) and the UCL (3.5) except sample 3.5, but it dosen't trail beyond the limits of either chart.

If the Ho in an ANOVA test is rejected, the next test you would want to run is?

Duncan multiple range test

ANOVA uses which distribution to determine the critical value?

F-distribution

If you have three data sets one with n1=15, n2=28, and n3=32 which test would you use to compare the data sets?

H test ir Kruskal-Wallis test

H test or Kruskal-Wallis test

Similar to the U-Test both are nonparametric tests and this test is an alternative to the one way analysis of variance. Used to compare 3 or more non-normally distributed data sets, the critical value comes from the x² distribution. DOF: k -1 where k is the # of data sets. R_i is the sum of all ranks for the ith sample n_i is the # of observations in the ith sample Dunn test is analogous to the Duncan test. n = total values of all samples Testing if the populations are identical or not. Hypothesis tests: H₀: Populations are identical H_a: Populations are not identical Combine your data sets in ascending order identifying the rank of each value and which data set the value comes from. Then sum all the ranks for each data set and plug into our test statistic. Use the x² table to find your critical value as well as the rejection region approach.

Linear Regression

The statistical method for determining a linear relationship between 2 variables assuming normality and independence. You can make 2 inferences using: - An r² value which tells us what % of the variation in the data can be accounted for by the equation. The closer to 1.0 the better the fit. For physical experiments r² should be > 0.9 and for people experiments r² should be > 0.30. - T values for coefficients: 1. Using the parameter α the hypothesis test: H₀: α=0 and the H_a α≠0 Its a 2 sided critical value based on n-2 df. 2. Using the slope parameter β: Hypothesis test H₀: β=0 and H_a- β≠0. Its a 2 sided critical value based on n-2 df

Duncan Multiple Range Test

This test is run after the ANOVA H₀ has been rejected once we know that one or more means are statistically different. This test tells us which means are statistically different from the others. It uses a least significance range value to determine differences for a set of means. If the actual range of means is larger than the last significant range value theres is a difference if the actual range of means is less than the last significant range value there is no difference. The range is underlined to represent all the data being the same. 1. Find the mean of each data set 2. Build your table for ranges for 2, 3, 4 etc adjacent means... starting with the largest # of adjacent means. The difference between the adjacent means will tell you the Actual ranges. 4. Calculate the Least significant ranges using the appropriate equation (MSE = Mean Square Error is usually given) 5.Compare your Actual Ranges to your least significant ranges starting with the largest # of adjacent means. Underline your data representing that the data is the same as you go.

A "good" value for r squared in a physical experiment is?

a. .25 b. .30 c. .70 d. .90 ANS: B. 30

Which of the values represents the change in the dependent variable for each additional unit of the independent variable in a prediction line.

a. Intercept b. Coefficient of the slope c. R squared d. P value ANS: Coefficient of the slope

Which of the values represents the degree to which the slope of the prediction line is statistically significant.

a. Intercept b. Coefficient of the slope c. R squared d. P value ANS: R squared value

Which of the following is most likely to be an independent variable in a regression model?

a. Sales of ice cream b. Number of incidents of violent crime c. Number of fleece jackets sold d. Outside temperature ANS: Outside Temperature

The critical value, with an H test is?

a. one sided b. two sided c. either a or b d. neither a nor b ANS: C. either a or b

The H test uses which distribution to determine the critical value?

x² distribution