Class 05. Using the Normal Intro to Descriptive Statistics EMBS Section 3.1, 3.2. and first part of 3.3

What we learned last class To love the NORMAL distribution? IT IS FLEXIBLE Parameter μ lets you center it anywhere. Parameter σ lets you specify the width (variance, uncertainty) If X is highly variable, pick a large σ NORMDIST makes probability calculations easy. No need for converting x to z in order to use the table in the book. No need for NORMSINV[(x-μ)/σ]. It approximates the binomial when n is big Just set μ=n*p and σ = [n*p*(1-p)]^.5

You can’t make it skewed. You can’t change the 68/95/99 breakdown. EMBS Fig 6.4, p 249 You can’t make it skewed. You can’t change the 68/95/99 breakdown. It is a pdf (prob density function). The height is probability density. Areas under the curve are probabilities

If X~N(μ,σ) And Y=a+b*X, Then Y~N(a+b*μ,b*σ) Temp(C)~N(10,10) Temp(F)= 32 + (9/5) Temp(C) -4 14 32 50 68 86 104 Temp(F)~N(50,18) -20 -10 0 10 20 30 40

Case: Lorex Pharmaceuticals Lorex produces Linatol (liquid medicine for high blood pressure) Sold in 10-ounce bottles. 600*12 = 7,200 fill attempts per shift Linatol costs them $0.40 per ounce to make. Bottles filled to more than 10 ounces get sold for $186 per case of 12. Bottles under-filled get sold as seconds at a 20% discount ($148.8 per case)

Case: Lorex Pharmaceuticals The Filling Machine has an adjustable target. Test have shown that the fill amounts will be independent (across fill attempts) and Normally distributed μ = target σ = 0.16 fluid ounces In other words (symbols?) X ~ N(target,0.16) What target should they use?

P(underfilled bottle) Worksheets…. Lorex Worksheet Minimum required 10 $186 $148.80 Target P(underfilled bottle) Expected Good Cases Expected Bad Cases Total Cases Expected Revenue 10.0   600 10.1 10.2 10.3

Fill Test Data Normal(10.2,0.16)? EXHIBIT 2 LOREX PHARMACEUTICALS Filling Line Test Results with Target = 10.2 9.89 10.41 10.53 10.20 10.23 10.15 10.17 10.32 10.04 10.48 10.11 10.29 10.35 10.16 10.19 10.00 10.06 10.21 10.22 9.76 10.09 10.12 10.10 10.02 10.36 9.99 10.05 10.07 10.24 10.40 10.27 10.14 10.76 10.13 9.79 10.31 10.47 9.84 9.96 10.30 10.33 9.97 10.42 10.25 10.64 10.01 10.18 10.54 10.37 10.44 9.85 9.91 10.45 10.28 10.03 9.98

Fill Test Data Descriptive Statistics Summary Statistics Amount Mean 10.198 Standard Error 0.014 Median 10.190 Mode #N/A Standard Deviation 0.163 Sample Variance 0.026 Kurtosis 0.771 Skewness 0.245 Range 0.997 Minimum 9.758 Maximum 10.756 Sum 1468.542 Count 144

Fill Test Data Histogram Bin Frequency 9.758 1 9.841 2 9.925 3 10.008 10 10.091 17 10.174 33 10.257 36 10.340 14 10.423 16 10.506 7 10.590 10.673 More

Assignment for class 06 Find some numerical data Get it into excel in one column with a label at the top Data Analysis (File, Options, Addins, Go, check analysis ToolPak, okay) Descriptive Statistics Check “summary statistics” One sheet A brief description of the data and where you got it. A printout (or hand copy) of the summary statistics A brief answer to the question…could these data have come from a normal distribution…why or why not?

Completed and extended worksheet Lorex Worksheet Minimum 10 $186 $148.80 Target P(underfilled bottle) Expected Good Cases Expected Bad Cases Total Cases Expected Revenue Expected Cost Expected Contribution 10.0 0.5 300.00 600 $100,440 $28,800 $71,640 10.1 0.266 440.41 159.59 $105,663 $29,088 $76,575 10.2 0.106 536.61 63.39 $109,242 $29,376 $79,866 10.3 0.030 581.76 18.24 $110,922 $29,664 $81,258 10.4 0.006 596.27 3.73 $111,461 $29,952 $81,509 10.5 0.001 599.47 0.53 $111,580 $30,240 $81,340 10.6 0.000 599.95 0.05 $111,598 $30,528 $81,070