Statistics 371 Notes7 University of Wisconsin Madison NOTES 7 HYPOTHESIS TESTING FOR FROM A SINGLE SAMPLE Z TEST T TEST BOOTSTRAP TEST AND POWER USING SAMPLE INFORMATION TO MAKE INFERENCES ABOUT POPULATION PARAMETERS We often want to know what is true for a population of subjects When it is impossible to obtain a census and summarize the population directly we must rely on statistical inference In this inference section we continue to apply the probability and simulation ideas we studied previously We will be developing techniques to determine if a claim or conjecture about a parameter of our population of interest is supported or refuted by what is observed in our sample We will also consider the limitations of our statistical inference Statistical inference is the practice of making informed estimates of population parameters or relationships based on sample data Tests of statistical hypotheses hypothesis tests examine whether sample data support or contradict a claim about the true value of a parameter or relationship These tests begin with making a statement about a population parameter or relationship Estimation of parameters is obtaining an estimate of the unknown parameter this is typically accomplished with a confidence interval HYPOTHESIS TESTING OUTLINE Steps for testing a hypothesis about a parameter 1 Identify the population parameter of interest in the context of the problem we ll start with in this note set 2 State the null and alternative hypothesis 3 Check the distributional assumptions 4 State and compute the appropriate test statistic based on distributional assumptions 5 Evaluate the evidence based on the test statistic a Using the rejection region method b Using the p value method c State the conclusion in the context of the problem 1 Statistics 371 Notes7 University of Wisconsin Madison HYPOTHESIS TESTING STEP 1 IDENTIFY THE POPULATION PARAMETER OF INTEREST AND DESCRIBE IT IN THE CONTEXT OF THE PROBLEM A statistical hypothesis is a claim or assertion about the value of a single population parameter about the values of several population parameters or about the form of an entire probability distribution May want to test the hypothesis that 5 where is the true May want to test the hypothesis that p 5 where p is the true population average population proportion May want to test the hypothesis that 1 2 0 where 1 and 2 are the true population averages for populations 1 and 2 respectively In this note set we will focus on hypothesis tests about the value of the population mean or average Paint Thickness example a The paint thicknesses measured in mm for 16 blocks randomly chosen over the course of the day were measured The observed thicknesses are 2 25 2 77 2 54 2 32 2 03 2 66 2 99 2 04 2 53 3 03 2 74 2 94 3 03 2 27 3 22 2 23 Summary statistics x 2 599 s 0 3801 Do these data give enough evidence that the population average paint thickness is different from 2 50 mm If so we may decide to recalibrate the machine Step 1 The parameter of interest is the true population average paint thickness T 2 Statistics 371 Notes7 University of Wisconsin Madison HYPOTHESIS TESTING STEP 2 STATE THE NULL HYPOTHESIS AND ALTERNATIVE HYPOTHESIS The null hypothesis H 0 is the claim that is assumed to be true H 0 is assumed to be true unless the sample provides sufficient evidence to the contrary H 0 statements regarding a single parameter always include the equality e g H o G 30 If the sample does not offer strong evidence against the null we say we fail to reject the null hypothesis The alternative hypothesis H A is the claim that is contradictory to H 0 Often a research hypothesis offers a specific range of values for a parameter An alternative hypothesis can take one of three forms where 0 is referred to e g H A G 30 as the null value o H A 0 o H A 0 o H A 0 If the sample offers strong evidence against the null hypothesis we say we reject the null hypothesis Paint Thickness example b Suppose the device that applies paint to blocks has specifications that say the true average thickness of the paint is 2 50 mm The device operator periodically evaluates a sample of paint thicknesses to determine if there is evidence that the average paint thickness is different from 2 50 mm and if so he will recalibrate the machine State the null and alternative hypothesis Parameter of interest T denotes the true average paint thickness of paint applied to all blocks Null Hypothesis H 0 T 2 50 no recalibration necessary Alternative Hypothesis H A T 2 50 recalibration necessary Pumpkin example a Tierra is a pumpkin farmer The pumpkin seed company claims the average weight of their Bushkin pumpkins is 8 lbs with standard deviation 0 75 lbs Tierra thinks her pumpkins look larger than that and wonders if 3 Statistics 371 Notes7 University of Wisconsin Madison the mean weight of her Bushkin pumpkins is heavier than claimed by the seed company Identify the population parameter of interest and write the null and alternative hypothesis for this problem Parameter if interest Let the true average weight of all Buskin pumpkins H A 8 H 0 8 implicitH 0 8 lbs versus Proportion of Cars example A county official wonders if the proportion of cars in the county with modified emission control devices differs from the statewide proportion of 0 15 The parameter of interest if the true proportion of cars in the county with modified emission control devices and the null and alternative hypothesis are H 0 0 15 versus H A 0 15 EXPECTED ERRORS IN HYPOTHESIS TESTING Because we are using sample information to estimate a population parameter we may make an incorrect statistical decision through no computational error Paint Thickness example c Consider the device paint thickness hypotheses Null Hypothesis H 0 T 2 50 no recalibration necessary Alternative Hypothesis H A T 2 50 recalibration necessary Suppose that based on the sample data we have strong evidence against the null hypothesis reject the null hypothesis and we decide we need to recalibrate the machine o This is the correct decision when the null is wrong and alternative is correct o This is an incorrect decision type I error when the null hypothesis is true Our sample happened to be extreme We can fix this type I error and denote it as with P Type 1 Error Rate Suppose that based on the sample data we have weak evidence against the null hypothesis fail to reject the null hypothesis and we decide we do not need to recalibrate the machine o This is the correct decision when
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