Virtual COMSATS Inferential Statistics Lecture23 Ossam Chohan Assistant

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Virtual COMSATS Inferential Statistics Lecture-23 Ossam Chohan Assistant Professor CIIT Abbottabad 1

Virtual COMSATS Inferential Statistics Lecture-23 Ossam Chohan Assistant Professor CIIT Abbottabad 1

Recap of last lecture • In our last sessions, we worked on: – Test

Recap of last lecture • In our last sessions, we worked on: – Test for independence. – Fisher’s Exact test. – Test for homogeneity. – Goodness of fit. – Review of Chi Square. 2

Objective of lecture-23 • In this lecture, we will understand problems related to: –

Objective of lecture-23 • In this lecture, we will understand problems related to: – F Statistic. – Testing ratio of two variances. – Introduction to ANOVA. • One way ANOVA. • Two way ANOVA. – Multiple Comparison Test 3

The F Distribution • The F distribution is the ratio of two variance estimates:

The F Distribution • The F distribution is the ratio of two variance estimates: • Also the ratio of two chi-squares, each divided by its degrees of freedom: In our applications, v 2 will be larger than v 1 and v 2 will be larger than 2. In such a case, the mean of the F distribution (expected value) is v 2 /(v 2 -2). 4

F Distribution • F depends on two parameters: v 1 and v 2 (df

F Distribution • F depends on two parameters: v 1 and v 2 (df 1 and df 2). The shape of F changes with these. Range is 0 to infinity. Shaped a bit like chisquare. • F tables show critical values for df in the numerator and df in the denominator. • F tables are 1 -tailed; can figure 2 -tailed if you need to (but you usually don’t). 5

Two-Sample Test for Variances To compare population variances, and , use the F-distribution. Let

Two-Sample Test for Variances To compare population variances, and , use the F-distribution. Let s 12 and s 22 represent the sample variances of two different populations. If both populations are normal and the population variances, and , are equal, then the sampling distribution is called an F-distribution. s 12 always represents the larger of the two variances. 0. 8 0. 7 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 0. 0 d. f. N = 8 d. f. D = 20 0 1 2 3 4 5 6

F-Test for Variances To test whether variances of two normally distributed populations are equal,

F-Test for Variances To test whether variances of two normally distributed populations are equal, randomly select a sample from each population. Let s 12 and s 22 represent the sample variances where The test statistic is: The sampling distribution is an F distribution with numerator d. f. = n 1 – 1 and denominator d. f. = n 2 – 1. In F-tests for equal variances, only use the right tail critical value. For a right-tailed test, use the critical value corresponding to the one in the table for the given α. For a two-tail test, use the right-hand critical value corresponding to. 7

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Problem-23 Two independent samples of sizes 9 and 8 gave the sum of squares

Problem-23 Two independent samples of sizes 9 and 8 gave the sum of squares of deviations from their respective means as 160 and 91 respectively. Can the sample be regarded as drawn from the normal populations with equal variances? Given F 0. 05(8, 7)=3. 73; F 0. 05(7, 8)=3. 50. 1. Write the null and alternative hypothesis. H 0: δ 12=δ 22 H 1: δ 12≠δ 22 2. State the level of significance. 12

Problem-23 Solution 13

Problem-23 Solution 13

Assessment Problem-21 • A research was conducted to understand whether women have a greater

Assessment Problem-21 • A research was conducted to understand whether women have a greater variation in attitude on political issues than men. Two independent samples of 31 men and 41 women were used for study. The sample variances so calculated were 120 for women and 80 for men. Test whether the difference in attitude toward political issues is significant at 5% level of significance. 14

ANOVA-Overview 15

ANOVA-Overview 15

Role of Variation, Sum of square and etc… 16

Role of Variation, Sum of square and etc… 16

What is ANOVA? • ANOVA and t- tests do the same job. Is that

What is ANOVA? • ANOVA and t- tests do the same job. Is that so? – Both test for. . ? ? • Problem with multiple t-tests – Probability of making a Type 1 error. . – ANOVA avoids increased Type 1 error by doing 1 single test to compare 17

Analysis of Variance (ANOVA) • The purpose of ANOVA is same as the t

Analysis of Variance (ANOVA) • The purpose of ANOVA is same as the t tests we discussed in hypothesis testing chapter while comparing two populations. • The goal is to determine whether the mean differences between the populations are significant from which the samples were obtained. • The ANOVA can be used in situations where there are two or more means being compared as compare to t tests that are limited to situations where only two means are involved. 18

ANOVA Cont… • Analysis of variance is necessary to protect researchers from excessive risk

ANOVA Cont… • Analysis of variance is necessary to protect researchers from excessive risk of a Type I error in situations where a study is comparing more than two population means. • Although the methodology of several t test can be applied for comparing two populations at a time. • But in that case final alpha value can accumulate to be very large. 19

ANOVA Cont… • ANOVA provides the facility to evaluate all of the mean differences

ANOVA Cont… • ANOVA provides the facility to evaluate all of the mean differences (no matter how many distributions are there to compare) in a single hypothesis test using a single α-level and • This methodology keeps type-1 error under control. 20

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Total Variability Total variability can be divided into two categories: Between treatment variation Within

Total Variability Total variability can be divided into two categories: Between treatment variation Within Treatment variation (Error) Recall: Variance=Sum of squares/size or degree of freedom 22

ANOVA Discussion 23

ANOVA Discussion 23

Statistical Hypotheses for ANOVA • Goal of ANOVA is to decide between the null

Statistical Hypotheses for ANOVA • Goal of ANOVA is to decide between the null and alternative hypotheses: • Ho (null): There are no differences between the populations (treatments). – The observed differences. . – That is, in the population, detergent quality. . . – Ho: u 1 = ? ? ? 24

Statistical Hypotheses for ANOVA • H 1(alternative): The differences between the sample means represent.

Statistical Hypotheses for ANOVA • H 1(alternative): The differences between the sample means represent. . – That is, the populations (treatments) really are. . – E. g. detergents does. . – H 1: u 1 ≠ is one possibility – General form of H 1: At least one population mean is. . 25

Test statistic for Anova: The Numerator • ANOVA similar logic and structure to t-test:

Test statistic for Anova: The Numerator • ANOVA similar logic and structure to t-test: t = obtained difference between [the 2] sample means difference expected by chance (due to sampling error) • In ANOVA we calculate a F ratio rather. F =. . between sample means expected by chance (due to sampling error) 26

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Problem-24 • The three samples below have been obtained from the normal population with

Problem-24 • The three samples below have been obtained from the normal population with equal variances. Test the hypothesis at 5% level that the population means are equal. Sample-1 Sample-2 Sample-3 8 7 12 10 5 9 7 10 13 14 9 12 11 9 14 28

Problem-24 Solution 29

Problem-24 Solution 29

Problem-24 Solution 30

Problem-24 Solution 30