STA 291 Spring 2008 Lecture 17 Dustin Lueker

Significance Test 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 STA 291 Spring 2008 Lecture 17

Logical Procedure State a hypothesis that you would like to find evidence against Get data and calculate a statistic Sample mean Sample proportion Hypothesis determines the sampling distribution of our statistic If the sample value is very unreasonable given our initial hypothesis, then we conclude that the hypothesis is wrong STA 291 Spring 2008 Lecture 17

Elements of a Significance Test Assumptions Type of data, population distribution, sample size Hypotheses Null hypothesis H0 Alternative hypothesis H1 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 STA 291 Spring 2008 Lecture 17

P-value 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 STA 291 Spring 2008 Lecture 17

Conclusion 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 STA 291 Spring 2008 Lecture 17

P-values and Their Significance Highly significant “Overwhelming evidence” .01<p-value<.05 Significant “Strong evidence” .05<p-value<.1 Not Significant “Weak evidence p-value>.1 “No evidence” Whether or not a p-value is considered significant typically depends on the discipline that is conducting the study STA 291 Spring 2008 Lecture 17

Terminology 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? STA 291 Spring 2008 Lecture 17

Type I and Type II Errors Decision Reject H0 Do Not Reject H0 Condition of H0 True Type I Error Correct False Type II Error STA 291 Spring 2008 Lecture 17

Type I and Type II Errors 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 STA 291 Spring 2008 Lecture 17

Example 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 H0 instead of saying that we accept H0 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 Spring 2008 Lecture 17

How to choose α? 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 STA 291 Spring 2008 Lecture 17

How to choose α? Which area of study would be most likely to use a very small level of significance? Social Sciences Medical Physical Sciences STA 291 Spring 2008 Lecture 17

Hypotheses H0: μ=μ0 H1: μ≠μ0 μ0 is the value we are testing against 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) STA 291 Spring 2008 Lecture 17

Test Statistic 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 STA 291 Spring 2008 Lecture 17

Example The mean age at first marriage for married men in a New England community was 22 years in 1790 For a random sample of 40 married men in that community in 1990, the sample mean age at first marriage was 26, assume the population standard deviation is 9 State the hypotheses, find the test statistic and p-value for testing whether or not the mean has changed, interpret Make a decision, using a significance level of 5% STA 291 Spring 2008 Lecture 17

P-value Has the advantage that different test results from different tests can be compared Always a number between 0 and 1, no matter why 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 STA 291 Spring 2008 Lecture 17

One-Sided Significance Tests 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 μ0 We need a one-sided test with hypotheses Null hypothesis can also be written with an equals sign STA 291 Spring 2008 Lecture 17

Example For a large sample test of the hypothesis the z test statistic equals 1.04 Find the p-value and interpret Suppose z=2.5 rather than 1.04, find the p-value Does this provide stronger or weaker evidence against the null hypothesis? Now consider the one-sided alternative For one-sided tests, the calculation of the p-value is different “Everything at least as extreme as the observed value” is everything above the observed value in this case Notice the alternative hypothesis STA 291 Spring 2008 Lecture 17

One-Sided vs. Two-Sided Test Two sided tests are more common in practice Look for formulations like “test whether the mean has changed” “test whether the mean has increased” “test whether the mean is the same” “test whether the mean has decreased” STA 291 Spring 2008 Lecture 17

Example If someone wanted to test to see if the average miles a social worker drives in a month was at least 2000 miles, what would H1 be? H0? μ<2000 μ≤2000 μ≠2000 μ≥2000 μ>2000 μ=2000 STA 291 Spring 2008 Lecture 17