What is a Hypothesis?

Transcription

1 What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population mean Example: The mean monthly cell phone bill in this city is μ = $42 population proportion Example: The proportion of adults in this city with cell phones is π = 0.68 Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-1

2 The Null Hypothesis, H 0 States the claim or assertion to be tested Example: The average number of TV sets in U.S. Homes is equal to three ( ) H 0 : μ Is always about a population parameter, not about a sample statistic 3 H 0 : μ 3 : 3 H 0 Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-2

3 The Null Hypothesis, H 0 Begin with the assumption that the null hypothesis is true Similar to the notion of innocent until proven guilty (continued) Refers to the status quo or historical value Always contains =, or sign May or may not be rejected Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-3

4 The Alternative Hypothesis, H 1 Is the opposite of the null hypothesis e.g., The average number of TV sets in U.S. homes is not equal to 3 ( H 1 : μ 3 ) Challenges the status quo Never contains the =, or sign May or may not be proven Is generally the hypothesis that the researcher is trying to prove Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-4

5 The Hypothesis Testing Process Claim: The population mean age is 50. H 0 : μ = 50, H 1 : μ 50 Sample the population and find sample mean. Population Sample Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-5

6 The Hypothesis Testing Process (continued) Sampling Distribution of 20 If it is unlikely that you would get a sample mean of this value... μ = 50 If H 0 is true... When in fact this were the population mean... then you reject the null hypothesis that μ = 50. Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-6

7 The Test Statistic and Critical Values If the sample mean is close to the assumed population mean, the null hypothesis is not rejected. If the sample mean is far from the assumed population mean, the null hypothesis is rejected. How far is far enough to reject H 0? The critical value of a test statistic creates a line in the sand for decision making -- it answers the question of how far is far enough. Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-7

8 The Test Statistic and Critical Values Sampling Distribution of the test statistic Region of Rejection Region of Non-Rejection Region of Rejection Critical Values Too Far Away From Mean of Sampling Distribution Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-8

9 Possible Errors in Hypothesis Test Decision Making Type I Error Reject a true null hypothesis Considered a serious type of error The probability of a Type I Error is Called level of significance of the test Set by researcher in advance Type II Error Failure to reject false null hypothesis The probability of a Type II Error is β Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-9

10 Possible Errors in Hypothesis Test Decision Making (continued) Possible Hypothesis Test Outcomes Actual Situation Decision H 0 True H 0 False Do Not Reject H 0 Reject H 0 No Error Probability 1 - α Type I Error Probability α Type II Error Probability β No Error Probability 1 - β Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-10

11 Possible Errors in Hypothesis Test Decision Making The confidence coefficient (1-α) is the probability of not rejecting H 0 when it is true. The confidence level of a hypothesis test is (1-α)*100%. The power of a statistical test (1-β) is the probability of rejecting H 0 when it is false. (continued) Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-11

12 Type I & II Error Relationship Type I and Type II errors cannot happen at the same time A Type I error can only occur if H 0 is true A Type II error can only occur if H 0 is false If Type I error probability ( ) Type II error probability ( β ), then Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-12

13 Factors Affecting Type II Error All else equal, β when the difference between hypothesized parameter and its true value β when β when σ β when n Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-13

14 Level of Significance and the Rejection Region H 0 : μ = 3 H 1 : μ 3 a /2 Level of significance = a a /2 0 Critical values Rejection Region This is a two-tail test because there is a rejection region in both tails Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 9-14

15 2 Test of Independence Similar to the 2 test for equality of more than two proportions, but extends the concept to contingency tables with r rows and c columns H 0 : The two categorical variables are independent (i.e., there is no relationship between them) H 1 : The two categorical variables are dependent (i.e., there is a relationship between them) Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 12-15

16 Wilcoxon Rank-Sum Test: Hypothesis and Decision Rule M 1 = median of population 1; M 2 = median of population 2 Test statistic = T 1 (Sum of ranks from smaller sample) Two-Tail Test Left-Tail Test Right-Tail Test H 0 : M 1 = M 2 H 0 : M 1 M 2 H 0 : M 1 M 2 H 1 : M 1 M 2 H 1 : M 1 < M 2 H 1 : M 1 > M 2 Reject Do Not Reject Reject Reject Do Not Reject Do Not Reject Reject T 1L T 1U T 1L T 1U Reject H 0 if T 1 T 1L or if T 1 T 1U Reject H 0 if T 1 T 1L Reject H 0 if T 1 T 1U Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 12-16

17 Wilcoxon Rank-Sum Test: Small Sample Example Sample data are collected on the capacity rates (% of capacity) for two factories. Are the median operating rates for two factories the same? For factory A, the rates are 71, 82, 77, 94, 88 For factory B, the rates are 85, 82, 92, 97 Test for equality of the population medians at the 0.05 significance level Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 12-17

18 Ranked Capacit y values: Tie in 3 rd and 4 th places Wilcoxon Rank-Sum Test: Small Sample Example Capacity Rank Factory A Factory B Factory A Factory B Rank Sums: (continued) Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 12-18

19 Wilcoxon Signed Ranks Test A nonparametric test for two related populations Steps: 1. For each of n sample items, compute the difference, D i, between two measurements 2. Ignore + and signs and find the absolute values, D i 3. Omit zero differences, so sample size is n 4. Assign ranks R i from 1 to n (give average rank to ties) 5. Reassign + and signs to the ranks R i 6. Compute the Wilcoxon test statistic W as the sum of the positive ranks Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 12-19

20 Wilcoxon Signed Ranks Test Statistic The Wilcoxon signed ranks test statistic is the sum of the positive ranks: W n' i1 ( ) R i For small samples (n < 20), use Table E.9 for the critical value of W Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 12-20

21 Wilcoxon Signed Ranks Test Statistic For samples of n > 20, W is approximately normally distributed with μ W n'(n' 1) 4 σ W n'(n' 1)(2n' 1) 24 Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 12-21

22 Wilcoxon Signed Ranks Test The large sample Wilcoxon signed ranks Z test statistic is Z STAT n' (n' 1) W 4 n' (n' 1)(2n' 1) To test for no median difference in the paired values: H 0 : M D = 0 H 1 : M D 0 24 Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 12-22

23 Kruskal-Wallis Rank Test Tests the equality of more than 2 population medians Use when the normality assumption for oneway ANOVA is violated Assumptions: The samples are random and independent Variables have a continuous distribution The data can be ranked Populations have the same variability Populations have the same shape Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 12-23

24 Kruskal-Wallis Test Procedure Obtain rankings for each value In event of tie, each of the tied values gets the average rank Sum the rankings for data from each of the c groups Compute the H test statistic Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 12-24

25 Kruskal-Wallis Test Procedure The Kruskal-Wallis H-test statistic: (with c 1 degrees of freedom) (continued) H 12 n(n 1) T c 2 j j1 n j 3(n 1) where: n = sum of sample sizes in all groups c = Number of groups T j = Sum of ranks in the j th group n j = Number of values in the j th group (j = 1, 2,, c) Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 12-25

26 Correlation vs. Regression A scatter plot can be used to show the relationship between two variables Correlation analysis is used to measure the strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the relationship No causal effect is implied with correlation Scatter plots were first presented in Ch. 2 Correlation was first presented in Ch. 3 Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-26

27 Introduction to Regression Analysis Regression analysis is used to: Predict the value of a dependent variable based on the value of at least one independent variable Explain the impact of changes in an independent variable on the dependent variable Dependent variable: the variable we wish to predict or explain Independent variable: the variable used to predict or explain the dependent variable Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-27

28 Simple Linear Regression Model Only one independent variable, Relationship between and Y is described by a linear function Changes in Y are assumed to be related to changes in Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-28

29 Types of Relationships Linear relationships Curvilinear relationships Y Y Y Y Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-29

30 Types of Relationships (continued) Strong relationships Weak relationships Y Y Y Y Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-30

31 Types of Relationships No relationship (continued) Y Y Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-31

32 Simple Linear Regression Model Dependent Variable Y Population Y intercept i β 0 Population Slope Coefficient β 1 Independent Variable i ε i Random Error term Linear component Random Error component Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-32

33 Simple Linear Regression Model (continued) Y Y i β 0 β 1 i ε i Observed Value of Y for i Predicted Value of Y for i Intercept = β 0 ε i Random Error for this i value Slope = β 1 i Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-33

34 Simple Linear Regression Equation (Prediction Line) The simple linear regression equation provides an estimate of the population regression line Estimated (or predicted) Y value for observation i Estimate of the regression intercept Estimate of the regression slope Ŷ i b 0 b 1 i Value of for observation i Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-34

35 Measures of Variation (continued) SST = total sum of squares (Total Variation) Measures the variation of the Y i values around their mean Y SSR = regression sum of squares (Explained Variation) Variation attributable to the relationship between and Y SSE = error sum of squares (Unexplained Variation) Variation in Y attributable to factors other than Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-35

36 Y Y i _ Y Y Measures of Variation SSE = (Y i - _ Y i ) 2 SST = (Y i - Y) 2 SSR = (Y _ - Y) 2 i (continued) Y _ Y i Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-36

37 Coefficient of Determination, r 2 The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable The coefficient of determination is also called r-squared and is denoted as r 2 r 2 SSR SST regression sum total sum of of squares squares note: 0 r 2 1 Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-37

38 Examples of Approximate r 2 Values Y r 2 = 1 Y r 2 = 1 r 2 = 1 Perfect linear relationship between and Y: 100% of the variation in Y is explained by variation in Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-38

39 Examples of Approximate r 2 Values Y Y 0 < r 2 < 1 Weaker linear relationships between and Y: Some but not all of the variation in Y is explained by variation in Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-39

40 Examples of Approximate r 2 Values Y r 2 = 0 No linear relationship between and Y: r 2 = 0 The value of Y does not depend on. (None of the variation in Y is explained by variation in ) Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 13-40