Week2AMTB2304 - Dr. Suren Phansalker ADM2304 MiniTab...

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Dr. Suren Phansalker ADM2304 MiniTab Approach to Confidence Intervals & Hypothesis Tests For Single Population Mean Some Basic Theory: (This is Strictly Optional!) When σ = σ (X) is known, {This is not the usual case} () ~( ) , ( ) X XNE X X n σ μσ ⎛⎞ == ⎜⎟ ⎝⎠ /2 1 X Pz z X αα μ α −< < = (1 ) { } ) ) 1 X z X −− > >− =− 1 X z X >> = () 1 PX z X X z X +> > = , or X X < + = This is the CI: X zX X z σμ < + X This is some times written as: CI: X Xz X Xz n ±= ± When σ = σ (X) is not known and must be estimated by calculating, s = s(X) 1 ) , ( ) n sX s Xt E X s X nn = {This is the usual case} ) )1 X Pt n t n < < = ) { } ) ( 1 ) ) ) 1 X Pt n t n −> − =− ) X n t n > ) ( ) ) ( PX t n X t n +− > > = , or ) ( ) ) ( n n < < = This is the CI: ) ( ) ) ( ) X tn s X X s X < < , some times also written as CI: ) ( ) ) ) s Xt n sX Xt n Xt n ±− = = 1
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An Example: A tire manufacturer tests tires to find their life in thousands of km. The data, KMTireK is stored in C1 and is shown below. 1. Print the Data: MTB > print c1 Data Display KMTireK 96 99 84 111 87 90 75 81 117 120 78 78 81 90 75 90 93 87 72 108 75 111 111 60 66 105 69 84 90 108 120 123 2. Describe the Data: MTB > desc c1; # Obtain the Basic Statistics for Data in C1 SUBC> gnhisto. # Also Draw the Histogram superposed with the Normal Curve Descriptive Statistics Variable N Mean Median TrMean StDev SE Mean KMTireK 32 91.69 90.00 91.61 17.33 3.06 Variable Minimum Maximum Q1 Q3 KMTireK 60.00 123.00 78.00 108.00 60 70 80 90 100 110 120 130 0 1 2 3 4 5 KMTireK Frequency Histogram of KMTireK, with Normal Curve 2
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3. Find NScore and Plot It: MTB > nscore c1 c2 # Find the NScore of C1 and put in C2 MTB > plot c2*c1 # Plot the NScore vs the Data in c1 120 110 100 90 80 70 60 2 1 0 -1 -2 KMTireK NScore It is “Reasonably” Normally distributed. 4. Find the Confidence Interval (t-interval) with Different “CCs”: MTB > TInterval 95.0 'KMTireK';# t-Confidence Interval of Data in C1 with Default CC of 95% SUBC> GBoxplot. # Also draw the Boxplot T Confidence Intervals Variable N Mean StDev SE Mean 95.0 % CI KMTireK 32 91.69 17.33 3.06 ( 85.44, 97.93) 0.4 0.3 0.2 0.1 0.0 X Density -2.0391 0.950 2.0391 0 Distribution Plot T, df=31 3
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Here are the actual Calculations by using the theory. Since CC = 1 – α = 0.95, LS = α = 0.05 t-Confidence Interval: /2 0.025 ( 1) (31) 2.0391 tn t α == 17.33 : ( 1) ( ) 91.69 2.0391( ) (85.4431,97.9369) 32 CI X t n s X ±− = 120 110 100 90 80 70 60 KMTireK Boxplot of KMTireK (with 95% t-confidence interval for the mean) [] X _ MTB > TInterval 99.0 C1; # t-CI of Data in C1 with 99% of CC specified SUBC> GBoxplot. # Also Draw the Boxplot T Confidence Intervals Variable N Mean StDev SE Mean 99.0 % CI KMTireK 32 91.69 17.33 3.06 ( 83.28, 100.09) Since CC = 1 – α = 0.99, LS = α = 0.01 t-Confidence Interval: / 2 0.005 ( 2.7440 t 17.33 : ( 1) ( ) 91.69 2.744( ) (83.2835,100.0965) 32 CI X t n s X = 120 110 100 90 80 70 60 KMTireK Boxplot of KMTireK (with 99% t-confidence interval for the mean) X _ 4
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MTB > TInt 99.5 C1; # t-CI of Data in C1 with 99% of CC specified SUBC> GBox. # Also Draw the Boxplot
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Week2AMTB2304 - Dr. Suren Phansalker ADM2304 MiniTab...

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