Lecture 16 Dustin Lueker

 Charlie claims that the average commute of his coworkers is 15 miles. Stu believes it is greater than that so he decides to ask some of his coworkers what their commute is. He asks 36 of them and finds that their average commute is miles with a standard deviation of 6 miles. ◦ Does this prove that Stu is correct and the average commute is greater than 15 miles?  If not how could you explain the sample mean being greater than 15 if the true, population mean (all the coworkers) isn’t? ◦ Can we use anything we have already learned to investigate this further? STA 291 Summer 2010 Lecture 162

 A way of statistically testing a hypothesis by comparing the data to values predicted by the hypothesis ◦ Data that fall far from the predicted values provide evidence against the hypothesis 3STA 291 Summer 2010 Lecture 16

1. State a hypothesis that you would like to find evidence against 2. Get data and calculate a statistic 1.Sample mean 2.Sample proportion 3. Hypothesis determines the sampling distribution of our statistic 4. If the sample value is very unreasonable given our initial hypothesis, then we conclude that the hypothesis is wrong 4STA 291 Summer 2010 Lecture 16

 H 0 : μ=μ 0 ◦ μ 0 is the value we are testing against  H 1 : μ≠μ 0 ◦ Most common alternative hypothesis  This is called a two-sided hypothesis since it includes values falling on two sides of the null hypothesis (above and below) 5STA 291 Summer 2010 Lecture 16

 The research hypothesis is usually the alternative hypothesis ◦ The alternative is the hypothesis that we want to prove by rejecting the null hypothesis  Assume that we want to prove that μ is larger than a particular number μ 0 ◦ We need a one-sided test with hypotheses  Null hypothesis can also be written with an equal sign 6STA 291 Summer 2010 Lecture 16

 Assumptions ◦ Type of data, population distribution, sample size  Hypotheses ◦ Null hypothesis  H 0 ◦ Alternative hypothesis  H 1  Test Statistic ◦ Compares point estimate to parameter value under the null hypothesis  P-value ◦ Uses the sampling distribution to quantify evidence against null hypothesis ◦ Small p-value is more contradictory  Conclusion ◦ Report p-value ◦ Make formal rejection decision (optional)  Useful for those that are not familiar with hypothesis testing 7STA 291 Summer 2010 Lecture 16

 The z-score has a standard normal distribution ◦ The z-score measures how many estimated standard errors the sample mean falls from the hypothesized population mean  The farther the sample mean falls from the larger the absolute value of the z test statistic, and the stronger the evidence against the null hypothesis 8STA 291 Summer 2010 Lecture 16

 How unusual is the observed test statistic when the null hypothesis is assumed true? ◦ The p-value is the probability, assuming that the null hypothesis is true, that the test statistic takes values at least as contradictory to the null hypothesis as the value actually observed  The smaller the p-value, the more strongly the data contradicts the null hypothesis 9STA 291 Summer 2010 Lecture 16

 Has the advantage that different test results from different tests can be compared ◦ Always a number between 0 and 1, no matter what type of data is being examined  Probability that a standard normal distribution takes values more extreme than the observed z-score  The smaller the p-value, the stronger the evidence against the null hypothesis and in favor of the alternative hypothesis 10STA 291 Summer 2010 Lecture 16

 In addition to reporting the p-value, sometimes a formal decision is made about rejecting or not rejecting the null hypothesis ◦ Most studies require small p-values like p<.05 or p<.01 as significant evidence against the null hypothesis  “The results are significant at the 5% level”  α=.05 11STA 291 Summer 2010 Lecture 16

 Charlie claims that the average commute of his coworkers is 15 miles. Stu believes it is greater than that so he decides to ask some of his coworkers what their commute is. He asks 36 of them and finds that their average commute is miles with a standard deviation of 6 miles. ◦ Construct a hypothesis test to see if Stu is correct using the P-Value method with a 5% level of significance STA 291 Summer 2010 Lecture 1612

 p-value<.01 ◦ Highly significant  “Overwhelming evidence” .01<p-value<.05 ◦ Significant  “Strong evidence” .05<p-value<.1 ◦ Not Significant  “Weak evidence  p-value>.1 ◦ Not Significant  “No evidence”  Whether or not a p-value is considered significant typically depends on the discipline that is conducting the study 13STA 291 Summer 2010 Lecture 16

 Significance level ◦ Alpha level  α  Number such that one rejects the null hypothesis if the p-values is less than it  Most common are.05 and.01 ◦ Needs to be chosen before analyzing the data  Why? 14STA 291 Summer 2010 Lecture 16

15 Decision Reject H 0 Do Not Reject H 0 Condition of H 0 True Type I Error Correct False Correct Type II Error STA 291 Summer 2010 Lecture 16

 α=probability of Type I error  β=probability of Type II error  Power=1-β ◦ The smaller the probability of Type I error, the larger the probability of Type II error and the smaller the power  If you ask for very strong evidence to reject the null hypothesis (very small α), it is more likely that you fail to detect a real difference  In reality, α is specified, and the probability of Type II error could be calculated, but the calculations are often difficult 16STA 291 Summer 2010 Lecture 16

 In a criminal trial someone is assumed innocent until proven guilty ◦ What type of error (in terms of hypothesis testing) would be made if an innocent person is found guilty? ◦ What type of error would be made if a guilty person is found not guilty? ◦ What does the Power represent (1-β)?  Also, the reason we only do not reject H 0 instead of saying that we accept H 0 is because of the way our hypothesis tests are set up  Just like in a criminal trial someone is found not guilty instead of innocent STA 291 Summer 2010 Lecture 1617

 If the consequences of a Type I error are very serious, then α should be small ◦ Criminal trial example  In exploratory research, often a larger probability of Type I error is acceptable  If the sample size increases, both error probabilities decrease 18STA 291 Summer 2010 Lecture 16

 Which area of study would be most likely to use a very small level of significance? ◦ Social Sciences ◦ Medical ◦ Physical Sciences STA 291 Summer 2010 Lecture 1619