Chapter 9 Introduction to Hypothesis Testing

Transcription

1 Chapter 9 Introduction to Hypothesis Testing Hypothesis Testing Hypothesis testing is an eample of inferential statistics We use sample information to draw conclusions about the population from which we sample We can never be 1% certain about the conclusions we draw about a population based upon one sample since a different sample selected from the same population could possibly lead one to draw a different conclusion about the same population. It is necessary to use probability to help eplain how reliable our conclusions are. We always assume that the sample data was selected randomly from the population Null and Alternative Hypotheses We begin a hypothesis test by stating the claims of the statistical test. There are two hypotheses that are formulated: Null Hypothesis (denoted by H o ) is the statement being tested which states a value for a population parameter and is initially assumed to be true. It is formulated for the sole purpose of trying to reject it by seeking evidence against it. Alternative Hypothesis (denoted by H a ) is a statement about the value of a population parameter which is different from the value stated in the null hypothesis. This hypothesis is suspected to be true and is supported by the rejection of the null hypothesis. Eample: State the null and alternative hypotheses for the following research A college newspaper claims that full-time college students work an average of 2 hours a week. A marketing professor who believes this claim is too high decides to conduct a study to test the newspaper s claim. A directional alternative hypothesis considers only one specified direction of difference away from the value stated in the null hypothesis and uses word equivalent to less than or greater than or too low or has increased A nondirectional alternative hypothesis considers both directions away from the value stated in the null hypothesis and uses word equivalent to is not or not equal to or different than Hypothesis Testing Procedure 1. Formulate the hypotheses 2. Determine the model to test the null hypothesis 3. Formulate a decision rule 4. Analyze the sample data 5. State the conclusion 1

2 There are two types of errors a researcher can make when performing a hypothesis test: Type I error: made when we reject the null hypothesis when it is true Type II error: made when we fail to reject the null hypothesis when it is false Possible Outcomes to a Statistical Hypothesis Test Reality: Conclusion: H is True H is False Reject H Type I Error Correct Decision Fail to Reject H Correct Decision Type II Error The development of a decision rule The objective of a hypothesis test is to use sample data to decide if one should reject or fail to reject the null hypothesis. Researchers use a predetermined decision rule to decide whether the sample data supports or refutes the null hypothesis. To decide whether to reject H o or fail to reject H o, the researcher compares the statistical result to the value(s) of the decision rule, referred to as the critical value(s). The level of significance (denoted by the Greek letter alpha, α) is the probability of a statistical result falling beyond the critical value, assuming that H o is true. It is also the probability of committing a type I error. The level of significance,, is usually given to you in the form of 1% or 5%. The statistical result computed from the sample that we are going to compare to the critical value is called the test statistic. The test statistic is computed from the sample information and is used to decide whether to reject or fail to reject the null hypothesis. It is the z or t score calculated for the sample mean. A one-tailed test (denoted by 1TT) is conducted when the alternative hypothesis is directional A two-tailed test (denoted by 2TT) is conducted when the alternative hypothesis is nondirectional Procedure to Formulate the Decision Rule: 1. Determine if the alternative hypothesis is directional or nondirectional and if it is a one tailed or two tailed test. 2. Identify the level of significance 3. Construct the appropriate hypothesis test model 4. Find the critical z score(s) for the corresponding level of significance. 5. State the decision rule 2

3 Formulating the Decision Rule Using the 5% Significance Level Type of Alternative Hypothesis, H a (less than) < (greater than) > Nondirectional (not equal to) Type of Test Hypothesis Test Model Two-tailed (2TT) Critical Z score(s) z c = 1.65 z c = 1.65 z LC = 1.96 and z RC = 1.96 Decision Rules statistic is less than 1.65 statistic is greater than 1.65 statistic is either less than 1.96 or greater than 1.96 Formulating the Decision Rule Using the 1% Significance Level Type of Alternative Hypothesis, H a (less than) < (greater than) > Nondirectional (not equal to) Type of Test Hypothesis Test Model Two-tailed (2TT) Critical Z score(s) z c = 2.33 z c = 2.33 z LC = 2.58 and z RC = 2.58 Decision Rules statistic is less than 2.33 statistic is greater than 2.33 statistic is either less than 2.58 or greater than

4 Chapter 1 Hypothesis Testing Involving a Population Mean Step 1: Formulate the two hypotheses H o and H a. Step 2: Determine the model to test the null hypothesis, H o. Under the assumption H o is true, the epected results are: a) The distribution of the test statistic is the hypothesis testing model and is either: i. Approimately a normal distribution, if the standard deviation of the population is known and the sample size is greater than 3, or ii. A t Distribution with degrees of freedom: df n 1 if the standard deviation of the population is unknown. b) The mean of the sampling distribution of the mean is given by the formula: o c) For a normal distribution: the standard error of the mean, denoted by is: n or For a t Distribution: the estimate of the standard error of the mean, written s, s is given by the formula: s n Step 3: Formulate the decision rule a) Alternate hypothesis: directional or non-directional b) Type of test: 1TT or 2TT c) Significance level: = 1% or = 5% d) Construct the hypothesis test model: find critical z score(s) z C, or zlc and z RC e) State the decision rule: reject the null hypothesis if the test statistic is Step 4: Analyze the sample data For a normal distribution the test statistic is z. For a t distribution the test statistic is t. s Step 5: State the conclusion by comparing the test statistic to the critical score of the decision rule. a) Reject H and accept Ha at or b) Fail to reject H at 4

5 Eamples from tetbook: Testing the hypothesis about a population mean, using information from a sample that is drawn from a population Eample 1.7 pg An automatic opening device for parachutes has a stated mean release time of 1 seconds with a population standard deviation of 3 seconds. A local parachute club decides to test this claim against the alternative hypothesis that the release time is not 1 seconds. The club purchases 36 of these devices and finds that the mean release time is 8.5 seconds. Does this sample result indicate that the opening device does not have a mean release time of 1 seconds? Use = 1%. 5

6 Eample 1.8 pg A tire manufacturer advertises that its brand of redial tires has a mean life of 4, miles with a population standard deviation of 1,5 miles. A consumer s research team decides to investigate this claim after receiving several complaints from people who believe this advertisement is false (i.e. the mean life of 4, miles is too high). If the research team tests 1 of these radial tires and obtains a sample mean tire life of 39,75, is the advertisement legitimate? Use = 5%. Homework for Chapter 1: Read handout and do pg # 19-29, 3, 32, 33, and 37 6