Applied Business Statistics Week 8 Hypothesis Testing Continued One tailed Two tailed t Agenda Today Hypothesis Testing Type I II Errors One sided or One tailed Two tailed Sigma unknown t Next Week Midterm review in lecture Midterm in class 3 8 3 6 Recap HYPOTHESIS TESTING From CI to HT Last Week Conf Intervals Used CI s to estimate unknown population parameters e g p We then made conclusions about the probability that the population parameter was contained within the interval we constructed using the sample with some degree of confidence typically 95 This Week Hypothesis Testing A more formal method for testing whether a given value is a reasonable value of a population parameter Purpose of HT to conclude whether the difference observed between our sample statistic and the hypothesized parameter is unlikely due to random sampling variation HT General Idea Procedure 1 2 3 General idea of a hypothesis test Making an initial assumption Collecting evidence data Based on the evidence data deciding whether or not to reject the initial assumption HT Components When writing a hypothesis test there are three things we need to know 1 2 3 The parameter that we are testing e g p The direction of the test two tailed right tailed or left tailed The value of the hypothesized parameter One Sided Tests for when is Known HYPOTHESIS TESTING Null and Alternative Hypotheses Researcher uses the sample data to evaluate the credibility of Ho The data will either provide support for or repute Ho Example Alternative Hypothesis H1 70 or Ha 70 If there is a big discrepancy between the data sample and the hypothesis we will reject the hypothesis Our strategy is to divide the normal distribution into two regions and examine high and low probability samples One sided hypothesis test right tailed because points to the right One Sided Test H0 H1 Points Right Values that differ significantly from Ho Type I Type II Error Test Result Ho True Ho False True State Ho True Correct Decision Type I Error Ho False Type II Error Correct Decision Alpha and Type I Error The alpha level is the probability of making a Type I error If the null hypothesis is correct only a small percentage of all possible sample means will fall within the critical region this percentage is determined by alpha If we want to reduce the probability of Type I error we reduce the alpha level if alpha 05 we have a 5 chance of being wrong but if we lower alpha to 01 we have only a 1 chance of being wrong Reducing alpha makes it harder to reject the null Type II Error Type II Error When a researcher fails to reject a null hypothesis that is actually false Usually means hypothesis test fails to detect an effect that is real It means you had bad luck in choosing a sample that makes your theory look incorrect even though it really is correct To reduce the probability of Type II error you need to increase the alpha level in other words make it a less stringent test making it easier to reject the null There is one way to simultaneously reduce the probability of both Type I and Type II errors increase the sample size Two tailed Tests for when is Known HYPOTHESIS TESTING Alpha probability that the sample mean you observe would be in this region tails of the distribution Total area outside the high probability region is 025 025 05 Thus the total area in the critical region 05 p 025 Tail If H0 were true p 025 Tail Alpha is set by the researcher e g 05 or 01 or 001 This is the probability of obtaining a sample this far away from the mean by chance Alpha 95 Testing for when is known Two tailed Q A new rail system is being built and financed on the expectation that it will service 8 500 passengers per day In the first 30 days of operation a daily average of 8 120 passengers used the rail system Using the 1 level of significance test whether the expectation was incorrect Assume the distribution of daily passengers is normally distributed with a of 950 Testing for when is known Two tailed Hypothesis Testing Steps State null and alternative hypotheses Calculate test statistic Consider decision rule State rejection decision Conclusion 1 2 3 4 5 1 State null H0 and alternative Ha or H1 hypotheses H0 8 500 H1 8 500 test whether the expectation was incorrect Testing for when is known Two tailed Hypothesis Testing Steps State null and alternative hypotheses Calculate test statistic Consider decision rule State rejection decision Conclusion 1 2 3 4 5 2 Calculate the test statistic 8 120 950 30 n 8 120 8 500 Testing for when is known Two tailed Hypothesis Testing Steps State null and alternative hypotheses Calculate test statistic Consider decision rule State rejection decision Conclusion 1 2 3 4 5 2 Calculate the test statistic 8 120 950 30 n Testing for when is known Two tailed Hypothesis Testing Steps State null and alternative hypotheses Calculate test statistic Consider decision rule State rejection decision Conclusion 1 2 3 4 5 3 Consider the decision rule 0 005 0 005 1 0 3 Consider the decision rule Testing for when is known Two tailed Hypothesis Testing Steps State null and alternative hypotheses Calculate test statistic Consider decision rule State rejection decision Conclusion 1 2 3 4 5 z crit 2 576 NORM S INV 0 995 0 005 2 576 1 0 005 0 2 576 Reject if z 2 576 or if z 2 576 3a Calculate p value 1 Testing for when is known Two tailed 3a Calculate p value The p value is the probability of getting a sample as extreme as ours given the null hypothesis is TRUE 0 0142 2 191 0 0 0142 2 191 p value NORM S DIST 2 191 TRUE 0 0142 3a Calculate p value 1 Testing for when is known Two tailed 3a Calculate p value The p value is the probability of getting a sample as extreme as ours given the null hypothesis is TRUE 0 0142 2 191 0 0 0142 2 191 p value NORM S DIST 2 191 TRUE 2 0 0285 Testing for when is known Two tailed Hypothesis Testing Steps State null and alternative hypotheses Calculate test statistic Consider decision rule State rejection decision Conclusion 1 2 3 4 5 4 State rejection decision Do Not Reject H0 H0 OR Do Not Reject At 1 level of significance as 2 576 z 2 576 z 2 191 OR At 1 level of significance as p 0 05 p 0 0285 5 Conclusion There is NOT enough evidence at the 1 level of significance to suggest that the daily average of passengers using the rail system is different from 8 500 Testing for when is known Two tailed Hypothesis Testing Steps State null and alternative hypotheses Calculate test statistic Consider decision rule State rejection decision