Transcript Chapter22
What is a Test of Significance? Statistical hypotheses – statements about population parameters Examples Mean weight of adult males is greater than 160 Proportion of students with a 4.0 GPA is less than .01 In statistics, we test one hypothesis against another The hypothesis that we want to prove is called the alternative hypothesis, H a Another hypothesis is formed that contradicts H a . This hypothesis is called the null hypothesis, H 0 After taking the sample, we must either: Reject H 0 and believe H a , or Fail to reject H 0 because there was not sufficient evidence to reject it (meaning there is not sufficient evidence to prove H a ) Types of errors H 0 is true H 0 is false Fail to reject H 0 Correct Type II error Reject H 0 Type I error Correct The probability with which we are willing to risk a type I error is called the level of significance of a test and is denoted The probability of making a type II error is denoted The quantity 1 is known as the power of a test. It represents the probability of rejecting H 0 when in fact it is false Decreasing increases Sample size is the only way to control both types of error Test Statistic – the statistic we compute to make the decision (sampling distribution of the test statistic must be known) The p-value of a hypothesis test is the smallest value of such that H 0 would have been rejected If p - value , reject H 0 If p - value , fail to reject H 0 Steps of a hypothesis test 1) State H a and H 0 2) Calculate the test statistic 3) Identify the p-value 4) Make decision and interpret results Example The current treatment for a type of cancer produces remission 20% of the time. An investigator wishes to prove that a new method is better. Suppose 26 of 100 patients go into remission using the new method. .05 There is not sufficient evidence to conclude the new method is better. Example Do less than 50% of people prefer Murray’s Vanilla Wafer’s when compared to other brands? Suppose that in a taste test 42 of the 250 choose Murray’s. .05 Conclude with 95% confidence that less than 50% of people prefer Murray’s Vanilla Wafer’s when compared to other brands. Inference about a Population Mean Remember x x n is the standard deviation of the sampling distribution which is referred to as the standard error Z x n has approximately a standard normal distribution ) Therefore, E Z ( n and the confidence interval is xE Example A sample of 100 visa accounts were studied for the amount of unpaid balance. x $645 and $132 Construct a 95% confidence interval We are 95% confident the mean unpaid balance of visa accounts is between $619.13 and $670.87. Construct a 99% confidence interval We are 99% confident the mean unpaid balance of visa accounts is between $611.00 and $679.00. Notice that as we increase the confidence level the interval gets wider Example A random sample of 500 apples yields Assume 1.1 oz. x 9.2 oz. Find a 95% confidence interval We are 95% confident the population mean weight of apples is between 9.104 and 9.296 oz. Example A consumer protection agency wants to prove that packages of Post Grape Nuts average less than 24 oz. .05 n 100 x 23.94 .13 Conclude with 95% confidence that packages of Post Grape Nuts has a mean less than 24 oz. Example It is desired to show the mean weight of a metal component is greater than 4.5 oz. .05 n 10 x 4.59 .504 There is not sufficient evidence to prove that the mean weight is greater than 4.5 oz.