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chapter 12 notes 0809.notebook March 30, 2012 Chapter 12 Significance Tests in Practice One sample mean and proportion hypothesis tests Skittle Quick Lab: It has been advertised that the color distribution for the original Skittles is 20% for each of the five colors. Since we have only developed tests for one sample statistics, lets test the company's claim of 20% for the green skittles. Ho: Ha : α= Conditions: Test Statistic: P-value = Conclusion: chapter 12 notes 0809.notebook March 30, 2012 Conducting Different Hypothesis Tests The structure of every hypothesis test will always be the same. What changes from test to test is the statistic and parameter that we will be using. The following are the different types of hypothesis tests that you will encounter: One-sample t test (mean) One-sample proportion test Two-sample mean test Two - sample proportion test χ2 Goodness of fit χ2 Test of Independence Linear Regression t- test (slope) Ex: Sweet cola Diet colas use artificial sweeteners to avoid sugar. These sweeteners gradually lose their sweetness over time. Manufacturers therefore test new colas for loss of sweetness before marketing them. Trained tasters sip the cola along with drinks of standard sweetness and score the cola on a "sweetness scale" of 1 to 10. The cola is then stored for a month at high temperature to imitate the effect of four months' storage at room temperature. Each taster scores the cola again after storage. Our data are the differences (score before storage minus score after storage) in the tasters' scores. The bigger these differences, the bigger the loss of sweetness. Here are the sweetness losses for a new cola, as measured by 10 trained tasters: 2.0 0.4 0.7 -0.4 2.2 -1.3 1.2 1.1 Are these data good evidence that the cola lost sweetness in storage? 2.3 2.0 chapter 12 notes 0809.notebook March 30, 2012 chapter 12 notes 0809.notebook March 30, 2012 Ex. 2: Diversify or be sued!!! An investor with a stock portfolio worth several hundred thousand dollars sued his broker because lack of diversification in his portfolio let to poor performance (low returns). The table below gives the rates of return for the 37 39 months that the account was managed by the broker. An arbitration panel compared these returns with average of the Standard & Poor's 500 stock index for the same period. Consider the 37 39 monthly returns as a random sample from the monthly returns the broker would generate if he managed the account forever. Are these returns compatible with a population mean of 0.95%, the S&P average? Monthly rates of return on a portfolio (%) -8.36 -2.35 -9.16 1.28 5.04 4.33 1.63 -3.58 -1.25 -0.65 -7.24 2.35 -2.27 6.13 -1.22 4.34 -2.14 -2.93 7.00 -10.27 12.22 -1.01 -9.14 -15.25 -5.11 -7.21 -1.41 -2.64 -8.66 -0.80 -0.09 12.03 Construct a 95% confidence interval for the above problem. Does your interval support your conclusion? PAIRED T-TEST This test is processed just like a one sample mean test because we only have one set of data that we are using. The sweetness of cola example is a paired test. It is paired by taster. Ex. Floral scents and learning 6.82 -1.03 -1.44 7.34 -2.56 chapter 12 notes 0809.notebook Ex. 1 March 30, 2012 chapter 12 notes 0809.notebook March 30, 2012 Confidence Intervals and Hypothesis Tests for Proportions: Ex.: Work Stress According to the National Institute for Occupational Safety and Health, job stress poses a major threat to the health of workers. A national survey of restaurant employees found that 75% said that work stress had a negative impact on their personal lives. A random sample of 100 employees from a large restaurant chain finds that 68 answer "Yes" when asked, "Does work stress have a negative impact on your personal life?" Is this good reason to think that the proportion of all employees in this chain who would say "Yes" differs from the national proportion? Construct a 95% CI for this situation. This is important!!!!! In the z-test when we are doing the standard deviation we use p, but in the confidence interval we use p-hat for the standard error. chapter 12 notes 0809.notebook March 30, 2012 Attachments SECTION 12.1 notes.doc