What’s going to be covered • Diagrams • Data Summary and Presentation • Binomial distribution • Engineering/Statistics Toolbox • Z-test • Type 2 Error • T-test • C2 Test
Dot Diagram Box Plot Q3 Q2 Q1 x = 1550 IQR 1.5 IQR 1050 1.5 IQR 1070 1030 1060 1080 1040 IQR = Inter Quartile Range
Data Summary Stem and Leaf Diagram Correlation Coefficient Quartile/Percentile Calculation QuartilePercentile Stem 1 2 3 4 5 7 8 Leaf 3 2 4 5 3 6 8 1 4 4 6 2 4 5 6 3 5 2 5 2 4 Freq. 1 3 5 7 4 0 1 n n n S S S (xi – x)(yi – y) (n + 1) 1st 2nd 3rd 5th 95th .05(n + 1) 4 i = 1 i = 1 i = 1 R = 2(n + 1) ) )( ( 4 .95(n + 1) (yi – y)2 (xi – x)2 3(n + 1) 4 Value will give ordered observation Interpolate as needed
Binomial Distribution ( ) n x px (1-p)n-x P(X = x) = We use Binomial Distribution when: Trials are independent Each trial results in one of two possible outcomes, success or failure The probability, p, remains constant n! ( ) n x = x!(n – x)!
Example 3-27 Samples of water have a 10% chance of containing high levels of organic solids. Assume the samples are independent with regards to the presence of the solids. Determine the probability that in the next 18 samples, exactly 2 contain high solids.
Solution X = the # of samples that contain high solids P= 0.1 N = 18 P(X=2) = P(X=2) =0.284
Engineering/Statistics Toolbox • Known as the procedure for hypothesis testing Steps for Generic Hypothesis Testing • 1. Identify Parameter Of Interest: • For instance; determine the saltiness of a potato chips • 2. State the Null Hypothesis (H0): • Standard that you are testing against, like the given average students test scores • 3. Alternative Hypothesis (H1): • Specify an appropriate alternative hypothesis • 4. Test Statistic • Equation you are going to use for each test. Z = X-m/(s/n^.5) • 6. Computations • Plug and chug • 7. Conclusion • Decide whether the Null Hypothesis should be rejected and report and that in the problem context.
Z-Test • When do you use it? • Known mean and known variance • Gives the probability density of when something is going to happen • Most of the time an alpha value will be given to you • If not, assume 0.05
Example Tom likes candy, his favorite is peanut butter cups. He’s been eating peanut butter cups everyday, and Tom thinks the peanut butter cup company is filling the bag with less peanut butter cups than they claim. He takes a sample of 8 bags and find the average amount of peanut butter cups per bag is 32 and they claim its 35. The standard deviation is 2.4. Are they filling the bags less, let α = 0.05.
solution Z = (x-µ)/(σ/) Z= -3.54 Reject the null hypothesis
Type II Error • When you fail to reject the null hypothesis when it is wrong then you have committed a type II error • b = f(Z0) • Power = 1 - b • For instance: Say you have a pop. of 50 beads with an average diameter of 10 mm (actual average diameter). However, your sample of 10 beads has an average of 15 mm. You want to confirm that a null hypothesis of 15 inches is correct. If you fail to reject the null you messed up.
T-Test • Unknown variance and known mean • You need to determine the sample variance • You need to know degrees of freedom • That will be n-1, (n is the sample size) • The same as the Z-test except with degrees of freedom and sample variance
Example 4-7 An experiment was performed in which 15 golf club drivers produced by a particular club maker were selected at random and their coefficients od restitution measured. It is of interest to determine if there is evidence (with α=0.05) to support a claim that the mean coefficient of restitution exceeds 0.82. n = 15. Observations X= 0.83725 S= 0.02456
Solution T = (x-µ)/(S/) T = 2.72 14 degrees of freedom P < 0.05 Reject null hypothesis, the mean coefficient of restitution exceeds 0.82.
C2-Test • This is a test on the sample variance • Much the same as T-test • Must know the sample variance, as well as the actual variance • This tests variance, NOT standard deviation
Example 4-10 A random sample of 20 liquid detergent bottles results in a sample variance of fill volume of s^2= 0.0153. if the variance of fill volume exceeds 0.01 an unacceptable proportion of bottles will be under filled and overfilled. Is there evidence in the sample data to suggest that the manufacturer has a problem with under and over filled bottles? α=0.05
Solution Significance of 0.05 and DOF=19, = 30.14 Fail to reject null, evidence is not strong enough to show the variance of fill volume exceeds 0.01.