HYPOTHESIS TESTING WHAT IS HYPOTHESIS TESTING? Hypothesis testing Based on sample evidence and probability theory Used to determine whether the hypothesis is a reasonable statement and should not be rejected, or is unreasonable and should be rejected A statistical hypothesis is an assertion or conjecture concerning one or more populations. What is a Hypothesis? Graduate school students sleep less than the average person. A certain peanut butter manufacturer is under filling jars so the mean content is less than 32 oz. STEPS IN TESTING HYPOTHESES 1. Establish hypotheses: state the null and alternative . 2. Select the level of significance ( 3. Determine the appropriate statistical and sampling distribution. 4. State the . 5. Gather sample data. Calculate the value of the test statistic. 6. Make a . 7. State the statistical . Make a managerial decision. 9-4 STEP 1: ESTABLISH HYPOTHESES: STATE THE NULL AND ALTERNATIVE . Null and alternate hypotheses Null Hypothesis H0 A statement about the value of a population parameter Alternative Hypothesis Ha: A statement that is accepted if the sample data provide evidence that the null hypothesis is false Three forms of the null and alternate hypotheses The null hypothesis always contains equality. H0: m = 0 Ha: m 0 H0: m1 = m2 Ha: m1 m2 H0: m = 0 Ha: m > 0 H0: m1 = m2 Ha: m1 > m2 H0: m = 0 Ha: m < 0 H0: m1 = m2 Ha: m1 < m2 Example Determine the null and alternative hypotheses in the following statements. 1. According to Giving and Volunteering in the United States, 2001 Edition, the mean charitable contribution per household in the US in 2000 was $1623. A researcher believes that the level of giving has changed since 2000. 2. Federal law requires that jars of peanut butter labeled as containing 32 oz. must contain at least 32 oz. A consumer advocate feels that a certain peanut butter manufacturer is underfilling jars so the mean contents are less than 32 oz. 3. According to the Centers for Disease Control and Prevention, 16% of children aged 6 to 11 years are overweight. A school nurse thinks the percentage of 6- to 11-year-olds who are overweight is higher in her district. STEP 2:. SELECT THE LEVEL OF SIGNIFICANCE ( Level of Significance. Null Hypothesis Ho is true Ho is false Researcher Accepts Rejects Ho Ho Correct Type I error decision (a Type II Correct Error Decision (power) (b RISK TABLE Notation: a = P(Type I Error) = P(rejecting Ho when Ho is true) b = P(Type II Error) = P(not rejecting H0 when Ha is true) 1- β = power of the test= P(rejecting Ho when Ho is false) Remark: the probability of making a Type I error, is called the level of significance. STEP 3: DETERMINE THE APPROPRIATE STATISTICAL AND SAMPLING DISTRIBUTION SOME TEST STATISTICS T-test Z-test F-test Chi-square 13 STUDENT'S T-TEST A test of the null hypothesis that the means of two normally distributed populations are equal. A test of whether the mean of a normally distributed population has a value specified in a null hypothesis. A test of whether the slope of a regression line differs significantly from 0. 14 Assumptions normal distribution of data (e.g. Wilk-Shapiro normality test, KS-test) equality of variances (F test, or more robust Levene's test) Samples may be independent or dependent, depending on the hypothesis and the type of samples: Independent samples are usually two, randomly selected groups Dependent samples are either two groups matched on some variable (for example, age) or are the same people being tested twice (called repeated measures) 15 Z-TEST The z value is based on the sampling distribution of X, which is normally distributed when the sample is reasonably large (recall Central Limit Theorem). data points should be independent from each other the distributions should be normal if n is low, if however n>30 the distribution of the data does not have to be normal the variances of the samples should be the same all individuals must be selected at random from the population 16 F TEST FOR TWO POPULATION VARIANCES S F S 2 1 2 2 dfnumerator 1 n1 1 dfdenom inator 2 n2 1 17 F-TEST FOR ONE WAY ANALYSIS OF VARIANCE (ANOVA) MSC F MSE ( x SST x n ( ( x ) x SSC SSb 2 2 2 2 i ni SSE SST SSC SSC MSC dfc n MSE SSE dfe STEP 4: . STATE THE DECISION RULE FOR ONE-TAILED TESTS Rejection Region Non Rejection Region m=12 oz Critical Value Reject Ho if Computed < - Critical Value Rejection Region y m=12 oz Critical Value Reject Ho if Computed > Critical value DECISION RULE FOR TWO-TAILED TESTS Rejection Region Rejection Region Non Rejection Region m=12 oz Critical Values Reject Ho if Computed > Critical Value 9-21 USING THE P-VALUE IN HYPOTHESIS TESTING p-Value Calculated from the probability distribution function or by computer The probability, assuming that the null hypothesis is true, of finding a value of the test statistic at least as extreme as the computed value for the test Decision Rule If the p-Value is larger than or equal to the significance level, a, H0 is not rejected. If the p-Value is smaller than the significance level, a, H0 is rejected. Step 6: Make a decision. MOVIE STEP 7: STATE THE STATISTICAL . STATING THE CONCLUSION OF A HYPOTHESIS TEST The sample evidence of a hypothesis testing situation enable us to decide whether to reject or not reject the null hypothesis. If we do not reject the null hypothesis, we are not saying the null hypothesis is true, only that it could be true. THANK YOU FOR LISTENING!