Lab 7 for Math 17 Hypothesis Testing I

Lab 7 for Math 17: Hypothesis Testing I 1 Chapter 21 - Short Notes Significance level is also called α level. This is the threshold you set for what a “small” p-value is. Common values are α = .01, .05, .10. Practical vs. Statistical significance. Types of Mistakes - Type I Error vs. Type II Error Power is the probability you correctly reject the null hypothesis. 2 True/False for Review + Hypothesis Steps 1. True/False. Decreasing sample size increases the power of a test. 2. True/False. A p-value is the probability the null hypothesis is true. 3. True/False. Null standard deviation in the one-sample z-test involves the value p̂, the sample proportion. 4. True/False. The significance level, α, in a hypothesis test is the probability of a Type II Error. 5. True/False. In a hypothesis test, α + β = 1. 6. True/False. When checking assumptions for a confidence interval for a proportion, one assumption is that np̂ and n(1 − p̂) are both ≥ 10. In a hypothesis test for a proportion, the related assumption is that np0 and n(1 − p0 ) are both ≥ 10. 7. True/False. When dealing with proportions, the population of responses (to the question of interest) can be viewed as a collection of zeros and ones. (Think about this, we haven’t discussed in class yet). 8. True/False. A p-value can be considered a conditional probability. 9. True/False. Practical significance is the same as statistical significance. 10. Briefly, list the steps in any hypothesis test. 1 3 Offenses in National Parks National Park rangers and police as well as other law enforcement officers responded to a total of 101,790 documented offenses in our national parks in 2002. (Data from Sourcebook of Criminal Justice Statistics.) A random sample of 10,000 of those offenses revealed that 420 offenses were serious (homicide, aggravated assault, motor vehicle theft, etc.). Is there significant evidence that the percentage of serious offenses in national parks was over 4 percent for the year 2002? Perform a hypothesis test, using α = .10. Interpret your test statistic. Interpret your p-value. Which type of error could have occurred in your hypothesis test? What is a possible consequence of this possible error? What would you have done differently, if anything, if you were told that out of the total of 101,790 offenses in 2002 exactly 3958 were serious? 2 4 CIs and Hypothesis Tests There is a relationship between confidence intervals and hypothesis tests. 1. A two-sided test at significance level α is equivalent to using a confidence interval at (1−α)*100% confidence and looking to see if the hypothesized value is in the confidence interval. If it is, you cannot reject the null hypothesis. If it isn’t, you can reject the null. 2. For a one-sided test at signficance level α, you need a confidence interval at (1 − 2α)*100% and again check for the hypothesized value in the confidence interval. If it is in the interval you cannot reject the null. In addition, the confidence interval must either be entirely below or above the hypothesized value (depending on situation) to reject the null. You can also obtain a 1-sided confidence interval (these are called confidence bounds) - Rcmdr does this automatically. 1. If you were doing a two-sided test at α = .1, what % confidence interval would you need? 2. For the test in 1. assume the appropriate CI was (.30,.42), would you be able to conclude that you could reject the null that the population proportion was .39? How about .45? 3. If you were doing a one-sided test at α = .01, what % confidence interval would you need? 4. For the test in 3. assume the appropriate CI was (.4,.43), and the alternative was greater than, would you reject the null that the population proportion was .39? How about .45? 5. If you were doing a one-sided test at α = .05 with a less than alternative and knew you were rejecting the null hypothesis that p = .4, what does that tell you about the corresponding 90% confidence interval? 6. If you were doing a one-sided test at α = .10 with a greater than alternative and knew you did not reject the null hypothesis that p = .7, what does that tell you about the corresponding 80% confidence interval? Could you conclude that you also would not reject the null hypothesis if you changed to a α = .05 level? 3 5 Mercury in Fish USGS information from this research is summarized here: http://water.usgs.gov/nawqa/mercury/mercury.handout.final.08172009.pdf and a full report is also available from the USGS. A 2009 study published by the U.S. Geological Survey collected several thousand fish from 291 different streams around the country from 1998 to 2005 to study mercury levels in fish. Every single fish sampled was found to be contaminated with mercury. However, 27 percent of sampled sites had mercury contamination levels above .3 ppm (which is the EPA cutoff for what is safe) and a total of 291 sites were sampled for this specific mercury data. Using this information, compute (and check conditions for) a 98 percent confidence level for the proportion of sites with mercury levels above the .3 ppm level. Now, suppose the USGS wanted to know if the proportion of sites with mercury levels above .3 ppm was signficantly higher than .25 and asked you to run a hypothesis test to help determine this. Set up your hypotheses. What significance level for your hypothesis test is equivalent to your 98% confidence interval above? Complete the hypothesis test (check conditions, etc.). Does the conclusion from your hypothesis test agree with your confidence interval? Explain. 4 6 Find the Mistakes A fellow student (not in class) is doing a hypothesis test for a population proportion for thesis work. However, the student is having trouble with some of the concepts. Here is some of the student’s work. What mistakes can you find? Help the student correct their work and interpretations. The setting is an analysis of Egyptian skulls. A previous analysis has shown that 25 percent of skulls from 150 AD, 200 BC, and 1850 BC had maximal skull breadth measurements of 132.25 mm or smaller. A new analysis is examining skulls from 4000 BC and 3300 BC (i.e. older skulls). 60 total skulls have been examined from these 2 older time periods. Your fellow student believes the skull measurements for the older skulls will be different due to interbreeding between groups that arrived in Egypt and the Egyptian population between these times and the newer times. Out of the 60 examined skulls, 35 skulls have maximal skull breadth measurements less than 132.25 mm. One sample t-test for population proportion. Let p be the population proportion of skulls with a maximal breadth measurement of 132.25 mm or smaller. (Parameter definition) H0 : p̂ = .25 and HA : p̂ = 35/60 = .5833 Assumptions: We assume we have a representative sample of skulls. np̂ = 60 ∗ .5833 and n(1 − p̂) = 60(.4167) are both greater than 10. Our test statistic is .25 − .5833 t= p = −5.96 .25 ∗ .75/60 which tells us that our observed sample proportion was 5.96 standard errors below the hypothesized proportion. p-value = P (Z < −5.96) which is approximately 0. This means our null hypothesis cannot be true. So, our decision is to accept the alternative hypothesis, and conclude that there is evidence that the proportion of older skulls with maximal skull breadth of 132.25 or smaller is different from .25. 5 7 To Turn In A poll conducted in late summer 2010 with a random sample of 3003 adults found that 61 percent of those surveyed opposed building the Muslim center near the Trade Center site. Assuming the appropriate conditions for inference are met, is this evidence that a majority of adults oppose building the center? Write a paragraph explaining your reasoning and showing any calculations you need to make for supporting work. 8 Note: New Vocab and Notation There is a LOT of new notation and vocabulary terms associated with hypothesis testing. You should be able to: identify and use appropriate notation for hypotheses differientiate between parameters and statistics differientiate between statistics and test statistics interpret test statistics interpret p-values interpret standard deviations and standard errors (null or regular versions) know when to reject the null hypothesis describe the two different types of mistakes in hypothesis testing explain what a significance level is explain what power is explain what the difference between statistical significance and practical significance is For confidence intervals, you should be able to: interpret standard errors interpret confidence intervals interpret confidence levels If you are having issues with ANY of these concepts, please come see me (can email for an office hour appointment outside regular office hours too!). The next 4 chapters are basically hypothesis testing and confidence intervals for 4 different parameters. 6