Pengujian Hipotesis Pertemuan 7 Matakuliah: D Statistika dan Aplikasinya Tahun: 2010

3 Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : 1.menjelaskan pengertian, jenis dan konsep-konsep dasar pengujian hipotesis 2.menerapkan pengujian hipotesis nilai tengah, proporsi dan ragam populasi Learning Outcomes

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Using Statistics The Concept of Hypothesis Testing Computing the p-value The Hypothesis Tests Testing population means, proportions and variances Pre-Test Decisions Hypothesis Testing

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., A hypothesis is a statement or assertion about the state of nature (about the true value of an unknown population parameter): The accused is innocent  = 100 Every hypothesis implies its contradiction or alternative: The accused is guilty  100 A hypothesis is either true or false, and you may fail to reject it or you may reject it on the basis of information: Trial testimony and evidence Sample data Using Statistics

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., One hypothesis is maintained to be true until a decision is made to reject it as false: Guilt is proven “beyond a reasonable doubt” The alternative is highly improbable A decision to fail to reject or reject a hypothesis may be: Correct A true hypothesis may not be rejected » An innocent defendant may be acquitted A false hypothesis may be rejected » A guilty defendant may be convicted Incorrect A true hypothesis may be rejected » An innocent defendant may be convicted A false hypothesis may not be rejected » A guilty defendant may be acquitted Decision-Making

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., A null hypothesis, denoted by H 0, is an assertion about one or more population parameters. This is the assertion we hold to be true until we have sufficient statistical evidence to conclude otherwise. H 0 :  = 100 The alternative hypothesis, denoted by H 1, is the assertion of all situations not covered by the null hypothesis. H 1 :  100 H 0 and H 1 are: Mutually exclusive –Only one can be true. Exhaustive –Together they cover all possibilities, so one or the other must be true. H 0 and H 1 are: Mutually exclusive –Only one can be true. Exhaustive –Together they cover all possibilities, so one or the other must be true. Statistical Hypothesis Testing

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Hypotheses about other parameters such as population proportions and and population variances are also possible. For example H 0 : p  40% H 1 : p < 40% H 0 :     H 1 :     Hypothesis about other Parameters

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The null hypothesis: Often represents the status quo situation or an existing belief. Is maintained, or held to be true, until a test leads to its rejection in favor of the alternative hypothesis. test statistic Is accepted as true or rejected as false on the basis of a consideration of a test statistic. The Null Hypothesis, H 0

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., test statistic A test statistic is a sample statistic computed from sample data. The value of the test statistic is used in determining whether or not we may reject the null hypothesis. decision rule The decision rule of a statistical hypothesis test is a rule that specifies the conditions under which the null hypothesis may be rejected. Consider H 0 :  = 100. We may have a decision rule that says: “Reject H 0 if the sample mean is less than 95 or more than 105.” In a courtroom we may say: “The accused is innocent until proven guilty beyond a reasonable doubt.” Consider H 0 :  = 100. We may have a decision rule that says: “Reject H 0 if the sample mean is less than 95 or more than 105.” In a courtroom we may say: “The accused is innocent until proven guilty beyond a reasonable doubt.” The Concepts of Hypothesis Testing

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., There are two possible states of nature: H 0 is true H 0 is false There are two possible decisions: Fail to reject H 0 as true Reject H 0 as false Decision Making

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., A decision may be correct in two ways: Fail to reject a true H 0 Reject a false H 0 A decision may be incorrect in two ways: Type I Error: Reject a true H 0 The Probability of a Type I error is denoted by . Type II Error: Fail to reject a false H 0 The Probability of a Type II error is denoted by . Decision Making

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., A decision may be incorrect in two ways: Type I Error: Reject a true H 0 The Probability of a Type I error is denoted by .  is called the level of significance of the test Type II Error: Accept a false H 0 The Probability of a Type II error is denoted by . 1 -  is called the power of the test.  and  are conditional probabilities: Errors in Hypothesis Testing

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., A contingency table illustrates the possible outcomes of a statistical hypothesis test. Type I and Type II Errors

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The p-value is the probability of obtaining a value of the test statistic as extreme as, or more extreme than, the actual value obtained, when the null hypothesis is true. The p-value is the smallest level of significance, , at which the null hypothesis may be rejected using the obtained value of the test statistic. Policy: Policy: When the p-value is less than , reject H 0. The p-value is the probability of obtaining a value of the test statistic as extreme as, or more extreme than, the actual value obtained, when the null hypothesis is true. The p-value is the smallest level of significance, , at which the null hypothesis may be rejected using the obtained value of the test statistic. Policy: Policy: When the p-value is less than , reject H 0. The p-Value NOTE: More detailed discussions about the p-value will be given later in the chapter when examples on hypothesis tests are presented.

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The power of a statistical hypothesis test is the probability of rejecting the null hypothesis when the null hypothesis is false. Power = (1 -  ) The power of a statistical hypothesis test is the probability of rejecting the null hypothesis when the null hypothesis is false. Power = (1 -  ) The Power of a Test

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., l The power depends on the distance between the value of the parameter under the null hypothesis and the true value of the parameter in question: the greater this distance, the greater the power. l The power depends on the population standard deviation: the smaller the population standard deviation, the greater the power. l The power depends on the sample size used: the larger the sample, the greater the power. The power depends on the level of significance of the test: the smaller the level of significance, , the smaller the power. Factors Affecting the Power Function

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., We will see the three different types of hypothesis tests, namely Tests of hypotheses about population means Tests of hypotheses about population proportions Tests of hypotheses about population proportions. We will see the three different types of hypothesis tests, namely Tests of hypotheses about population means Tests of hypotheses about population proportions Tests of hypotheses about population proportions. The Hypothesis Tests

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The tails of a statistical test are determined by the need for an action. If action is to be taken if a parameter is greater than some value a, then the alternative hypothesis is that the parameter is greater than a, and the test is a right-tailed test. H 0 :  50 H 1 :  50 The tails of a statistical test are determined by the need for an action. If action is to be taken if a parameter is greater than some value a, then the alternative hypothesis is that the parameter is greater than a, and the test is a right-tailed test. H 0 :  50 H 1 :  50 If action is to be taken if a parameter is less than some value a, then the alternative hypothesis is that the parameter is less than a, and the test is a left- tailed test. H 0 :  50 H 1 :  50 If action is to be taken if a parameter is less than some value a, then the alternative hypothesis is that the parameter is less than a, and the test is a left- tailed test. H 0 :  50 H 1 :  50 If action is to be taken if a parameter is either greater than or less than some value a, then the alternative hypothesis is that the parameter is not equal to a, and the test is a two-tailed test. H 0 :  50 H 1 :  50 If action is to be taken if a parameter is either greater than or less than some value a, then the alternative hypothesis is that the parameter is not equal to a, and the test is a two-tailed test. H 0 :  50 H 1 :  50 1-Tailed and 2-Tailed Tests

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Cases in which the test statistic is Z  is known and the population is normal.  is known and the sample size is at least 30. (The population need not be normal) Cases in which the test statistic is Z  is known and the population is normal.  is known and the sample size is at least 30. (The population need not be normal) Testing Population Means

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., An automatic bottling machine fills cola into two liter (2000 cc) bottles. A consumer advocate wants to test the null hypothesis that the average amount filled by the machine into a bottle is at least 2000 cc. A random sample of 40 bottles coming out of the machine was selected and the exact content of the selected bottles are recorded. The sample mean was cc. The population standard deviation is known from past experience to be 1.30 cc. Test the null hypothesis at the 5% significance level. H 0 :  2000 H 1 :  2000 n = 40 For  = 0.05, the critical value of z is The test statistic is: Do not reject H 0 if: [p-value  ] Reject H 0 if:  p-value  ] H 0 :  2000 H 1 :  2000 n = 40 For  = 0.05, the critical value of z is The test statistic is: Do not reject H 0 if: [p-value  ] Reject H 0 if:  p-value  ] Example : p-value approach

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Cases in which the test statistic is t  is unknown but the sample standard deviation is known and the population is normal. Cases in which the test statistic is t  is unknown but the sample standard deviation is known and the population is normal. Testing Population Means

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., According to the Japanese National Land Agency, average land prices in central Tokyo soared 49% in the first six months of An international real estate investment company wants to test this claim against the alternative that the average price did not rise by 49%, at a 0.01 level of significance. H 0 :  = 49 H 1 :  49 n = 18 For  = 0.01 and (18-1) = 17 df, critical values of t are ±2.898 The test statistic is: Do not reject H 0 if: [  t  2.898] Reject H 0 if: [t < ] or  t  2.898] H 0 :  = 49 H 1 :  49 n = 18 For  = 0.01 and (18-1) = 17 df, critical values of t are ±2.898 The test statistic is: Do not reject H 0 if: [  t  2.898] Reject H 0 if: [t < ] or  t  2.898] Additional Examples

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The rejection region of a statistical hypothesis test is the range of numbers that will lead us to reject the null hypothesis in case the test statistic falls within this range. The rejection region, also called the critical region, is defined by the critical points. The rejection region is defined so that, before the sampling takes place, our test statistic will have a probability  of falling within the rejection region if the null hypothesis is true. Rejection Region

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The nonrejection region is the range of values (also determined by the critical points) that will lead us not to reject the null hypothesis if the test statistic should fall within this region. The nonrejection region is designed so that, before the sampling takes place, our test statistic will have a probability 1-  of falling within the nonrejection region if the null hypothesis is true In a two-tailed test, the rejection region consists of the values in both tails of the sampling distribution. Nonrejection Region

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc.,  = x = 31.5 Population mean under H 0 95% confidence interval around observed sample mean It seems reasonable to reject the null hypothesis, H 0 :  = 28, since the hypothesized value lies outside the 95% confidence interval. If we’re 95% sure that the population mean is between and minutes, it’s very unlikely that the population mean is actually be 28 minutes. Note that the population mean may be 28 (the null hypothesis might be true), but then the observed sample mean, 31.5, would be a very unlikely occurrence. There’s still the small chance (  =.05) that we might reject the true null hypothesis. level of significance  represents the level of significance of the test. Picturing Hypothesis Testing

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., If the observed sample mean falls within the nonrejection region, then you fail to reject the null hypothesis as true. Construct a 95% nonrejection region around the hypothesized population mean, and compare it with the 95% confidence interval around the observed sample mean: x  % Confidence Interval around the Sample Mean  0 = % non- rejection region around the population Mean The nonrejection region and the confidence interval are the same width, but centered on different points. In this instance, the nonrejection region does not include the observed sample mean, and the confidence interval does not include the hypothesized population mean. Nonrejection Region

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., T If the null hypothesis were true, then the sampling distribution of the mean would look something like this: We will find 95% of the sampling distribution between the critical points and 28.98, and 2.5% below and 2.5% above (a two-tailed test). The 95% interval around the hypothesized mean defines the nonrejection region, with the remaining 5% in two rejection regions. Picturing the Nonrejection and Rejection Regions

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Construct a (1-  ) nonrejection region around the hypothesized population mean. Do not reject H 0 if the sample mean falls within the nonrejection region (between the critical points). Reject H 0 if the sample mean falls outside the nonrejection region. The Decision Rule

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., For testing hypotheses about population variances, the test statistic (chi-square) is: where is the claimed value of the population variance in the null hypothesis. The degrees of freedom for this chi-square random variable is (n – 1). Note: Since the chi-square table only provides the critical values, it cannot be used to calculate exact p-values. As in the case of the t-tables, only a range of possible values can be inferred. For testing hypotheses about population variances, the test statistic (chi-square) is: where is the claimed value of the population variance in the null hypothesis. The degrees of freedom for this chi-square random variable is (n – 1). Note: Since the chi-square table only provides the critical values, it cannot be used to calculate exact p-values. As in the case of the t-tables, only a range of possible values can be inferred. Testing Population Variances

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., A manufacturer of golf balls claims that they control the weights of the golf balls accurately so that the variance of the weights is not more than 1 mg 2. A random sample of 31 golf balls yields a sample variance of 1.62 mg 2. Is that sufficient evidence to reject the claim at an  of 5%? Example Let  2 denote the population variance. Then H 0 :  2  1 H 1 :  2  In the template (see next slide), enter 31 for the sample size and 1.62 for the sample variance. Enter the hypothesized value of 1 in cell D11. The p-value of appears in cell E13. Since This value is less than the  of 5%, we reject the null hypothesis. Let  2 denote the population variance. Then H 0 :  2  1 H 1 :  2  In the template (see next slide), enter 31 for the sample size and 1.62 for the sample variance. Enter the hypothesized value of 1 in cell D11. The p-value of appears in cell E13. Since This value is less than the  of 5%, we reject the null hypothesis.

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., While the null hypothesis is maintained to be true throughout a hypothesis test, until sample data lead to a rejection, the aim of a hypothesis test is often to disprove the null hypothesis in favor of the alternative hypothesis. This is because we can determine and regulate , the probability of a Type I error, making it as small as we desire, such as 0.01 or Thus, when we reject a null hypothesis, we have a high level of confidence in our decision, since we know there is a small probability that we have made an error. A given sample mean will not lead to a rejection of a null hypothesis unless it lies in outside the nonrejection region of the test. That is, the nonrejection region includes all sample means that are not significantly different, in a statistical sense, from the hypothesized mean. The rejection regions, in turn, define the values of sample means that are significantly different, in a statistical sense, from the hypothesized mean. Statistical Significance

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., When the p-value is smaller than 0.01, the result is called very significant. When the p-value is between 0.01 and 0.05, the result is called significant. When the p-value is between 0.05 and 0.10, the result is considered by some as marginally significant (and by most as not significant). When the p-value is greater than 0.10, the result is considered not significant. When the p-value is smaller than 0.01, the result is called very significant. When the p-value is between 0.01 and 0.05, the result is called significant. When the p-value is between 0.05 and 0.10, the result is considered by some as marginally significant (and by most as not significant). When the p-value is greater than 0.10, the result is considered not significant. The p-Value: Rules of Thumb

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., In a two-tailed test, we find the p-value by doubling the area in the tail of the distribution beyond the value of the test statistic. p-Value: Two-Tailed Tests z f ( z ) p-value=double the area to left of the test statistic =2(0.3446)=0.6892

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The further away in the tail of the distribution the test statistic falls, the smaller is the p-value and, hence, the more convinced we are that the null hypothesis is false and should be rejected. In a right-tailed test, the p-value is the area to the right of the test statistic if the test statistic is positive. In a left-tailed test, the p-value is the area to the left of the test statistic if the test statistic is negative. In a two-tailed test, the p-value is twice the area to the right of a positive test statistic or to the left of a negative test statistic. For a given level of significance,  : Reject the null hypothesis if and only if  p-value The further away in the tail of the distribution the test statistic falls, the smaller is the p-value and, hence, the more convinced we are that the null hypothesis is false and should be rejected. In a right-tailed test, the p-value is the area to the right of the test statistic if the test statistic is positive. In a left-tailed test, the p-value is the area to the left of the test statistic if the test statistic is negative. In a two-tailed test, the p-value is twice the area to the right of a positive test statistic or to the left of a negative test statistic. For a given level of significance,  : Reject the null hypothesis if and only if  p-value The p-Value and Hypothesis Testing

36 RINGKASAN Pengujian Hipotesis : Pengertian dan jenis hipotesis Konsep-konsep pengujian hipotesis Pengujian nilai tengah- ragam diketahui dan ragam tidak diketahui, ukuran sample kecil