PROBABILITY & STATISTICAL INFERENCE LECTURE 6 MSc in Computing (Data Analytics)
Lecture Outline Quick Recap Testing the difference between two sample means Practical Hypothesis Testing Analysis Of Variance
General Steps in Hypotheses testing 1. From the problem context, identify the parameter of interest. 2. State the null hypothesis, H Specify an appropriate alternative hypothesis, H Choose a significance level, . 5. Determine an appropriate test statistic. 6. State the rejection region for the statistic. 7. Compute any necessary sample quantities, substitute these into the equation for the test statistic, and compute that value. 8. Decide whether or not H 0 should be rejected and report that in the problem context.
Type of questions that can be answered with Two sample hypothesis tests A manufacturing plant want to compare the defective rate of items coming off two different process lines. Whether the test results of patients who received a drug are better than test results of those who received a placebo. The question being answered is whether there is a significant (or only random) difference in the average cycle time to deliver a pizza from Pizza Company A vs. Pizza Company B.
Difference in Means of Two Normal Distributions, Variances Known
Test Assumptions
Example
The P-Value is the exact significance level of a statistical test; that is the probability of obtaining a value of the test statistic that is at least as extreme as that when the null hypothesis is true
Confidence Interval on a Difference in Means, Variances Known
Example
Difference in Means of Two Normal Distributions, Variances unknown We wish to test: The pooled estimator of 2 :
Difference in Means of Two Normal Distributions, Variances unknown
Example
Confidence Interval on the Difference in Means, Variance Unknown
Example
Practical Hypothesis Testing 1. From the problem context, identify the parameter of interest. 2. State the null hypothesis, H Specify an appropriate alternative hypothesis, H Choose a significance level, . 5. Calculate the P-value using a software package of choice. 6. Decide whether or not H 0 should be rejected and report that in the problem context. Reject H 0 when P-Value is less than . (Golden rule: Reject H 0 for small )
Some Reserach Look up the correct formula for calculating the hypotheses test between two proportions What are the assumptions for the test Find an example of the research
Analysis of Variance
Introduction In the previous section we were concerned with the analysis of data where we compared the sample means. Frequently data contains more that two samples, they may compare several treatments. In this lecture we introduce statistical analysis that allows us compare the mean of more that two samples. The method is called ‘Analysis of Variance ‘ or AVOVA for short.
Total Sum of Squares Data set: 14, 12, 10, 6,4, 2 Group A: 6,4, 2 Group B: 14, 12, 10 Overall Mean : 8 Total Sum of Squares: SS T = (14-8) 2 + (12-8) 2 + (10-8) 2 + (6-8) 2 + (4-8) 2 + (2-8) 2 =112
Between Group Variation Sum of Squares of the Model: SS m = n a (µ - µ a ) 2 + n b (µ - µ b ) 2 =3*(8-4) 2 + 3*(8-12) 2 =96
Within Group Variation Sum of Squares of the Error: SS e = = (14-12) 2 + (12-12) 2 + (10-12) 2 + (6-4) 2 + (4- 4) 2 + (4-2) 2 + = 16
Structure of the Data GroupObservationTotalMean 1x 11 x x 1n x1x1 2x 21 x x 2n x2x ax a1 x a x an xaxa Total
ANOVA Table SourceDegrees of Freedom Sum Of SquaresMean Square F- Stat Modela - 1SS M /(a-1)MS M / MS E Errorn-a SS E /(n-a) Totaln-1 SS T /(n-1) Where : n is the sample size and a is the number of groups
ANOVA Table – Original Example SourceDegrees of Freedom Sum Of SquaresMean Square F- Stat Model2 - 1 = Error6 – 2 = Total6 – 1 = 5112 Where : n is the sample size and a is the number of groups
Model Assumptions Independence of observations within and between samples normality of sampling distribution equal variance - This is also called the homoscedasticity assumption
The ANOVA Equation We can describe the observations in the above table using the following equation: Where : n is the sample size and k is the number of groups
ANOVA Hypotheses We wish to test the hypotheses: The analysis of variance partitions the total variability into two parts.
Example
Graphical Display of Data Figure 13-1 (a) Box plots of hardwood concentration data. (b) Display of the model in Equation 13-1 for the completely randomized single-factor experiment
Example We can use ANOVA to test the hypotheses that different hardwood concentrations do not affect the mean tensile strength of the paper. The hypotheses are: The ANOVA table is below:
Example The p-value is less than 0.05 therefore the H 0 can be rejected and we can conclude that at least one of the hardwood concentrations affects the mean tensile strength of the paper.
Test Model Assumptions Use the Bartletts Test to test for homoscedasticity assumption Bartlett's test (Snedecor and Cochran, 1983) is used to test if k samples have equal variances. Bartlett's test is sensitive to departures from normality. That is, if your samples come from non- normal distributions, then Bartlett's test may simply be testing for non-normality. The Levene test is an alternative to the Bartlett test that is less sensitive to departures from normality.
Barlett Test for Equal Variance The hypotheses for the Barlett test are as follows: The barlett test statistic follows a chi-squared distribution Interpert the p-value like any other hypothese test
If the Assumption of Equal Variance is not met If the assumption for equal variance is not met use the Welches ANOVA Assignment for next week: Investigate the difference between the standard ANOVA and Welches ANOVA?
Demo
Confidence Interval about the mean For 20% hardwood, the resulting confidence interval on the mean is
Confidence Interval about on the difference of two treatments For the hardwood concentration example,
An Unbalanced Experiment
Multiple Comparisons Following the ANOVA The least significant difference (LSD) is If the sample sizes are different in each treatment:
Example: Multi-comparison Test
Demo
Exercises