Lecture 17 Dustin Lueker

 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 2STA 291 Summer 2010 Lecture 17

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 calculated value in 2 is very unreasonable given 3, then we conclude that the hypothesis is wrong 3STA 291 Summer 2010 Lecture 17

 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 4STA 291 Summer 2010 Lecture 17

 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 5STA 291 Summer 2010 Lecture 17

 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 6STA 291 Summer 2010 Lecture 17

 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 with a standard deviation of 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% 7STA 291 Summer 2010 Lecture 17

 For a large sample test of the hypothesis the z test statistic equals 1.04 ◦ Now consider the one-sided alternative  Find the p-value and interpret  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 8STA 291 Summer 2010 Lecture 17

 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 H 1 be? H 0 ? 1.μ< μ≤ μ≠ μ≥ μ> μ=2000 STA 291 Summer 2010 Lecture 179

 Which p-value would indicate the most significant evidence against the null hypothesis? STA 291 Summer 2010 Lecture 1710

 Range of values such that if the test statistic falls into that range, we decide to reject the null hypothesis in favor of the alternative hypothesis ◦ Type of test determines which tail(s) the rejection region is in  Left-tailed  Right-tailed  Two-tailed 11STA 291 Summer 2010 Lecture 17

 Find the rejection region for each set of hypotheses and levels of significance.  H 1 : μ > μ 0 ◦ α =.05  H 1 : μ < μ 0 ◦ α =.02  H 1 : μ ≠ μ 0 ◦ α =.01 STA 291 Summer 2010 Lecture 1712

 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 13STA 291 Summer 2010 Lecture 17

 Thirty-second commercials cost $2.3 million during the 2001 Super Bowl. A random sample of 116 people who watched the game were asked how many commercials they watched in their entirety. The sample had a mean of and a standard deviation of Can we conclude that the mean number of commercials watched is greater than 15?  State the hypotheses, find the test statistic and use the rejection region ◦ Make a decision, using a significance level of 5% 14STA 291 Summer 2010 Lecture 17