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-187642-299403Statistical Inference 020000Statistical Inference Section 9.1 – Intro to Significance Tests 276225112395 Confidence intervals are used to estimate a parameter (µ or p) based on a sample statistic (x or p). Significance Tests are used to evaluate a claim about a population parameter (µ or p) based on a sample statistic (x or p). ***Used to reject or fail to reject explanations on the basis of their relative likelihood. Hypotheses: The Null Hypothesis (H0) is a statement of “no difference.” This is the hypothesis we weigh evidence against. The Alternative Hypothesis (Ha) is the claim we are trying to find evidence for. Means Proportions 1304925126365 Null: H0: µ = ?0 or H0: p = p0 Alternative: 500248139370*indicated with greater than, more than, or increased *indicated with less than, or decreased *indicated with a change or difference 020000*indicated with greater than, more than, or increased *indicated with less than, or decreased *indicated with a change or difference 1304925698500 (one sided): Ha: µ > ?0 or Ha: p > p0 Ha: µ < ?0 or Ha: p < p0 1304925115570 (two sided) Ha: µ ? ?0 or Ha: p ? p0 Check Your Understanding, p. 541 For each of the following settings, (a) describe the parameter of interest, and (b) state appropriate hypotheses for a significance test. 1. According to the Web site sleepdeprivation.com, 85% of teens are getting less than eight hours of sleep a night. Janie wonders whether this result holds in her large high school. She asks an SRS of 100 students at the school how much sleep they get on a typical night. In all, 75 of the responders said less than 8 hours. (a) p = proportion of all students at Janie’s high school who get less than 8 hours of sleep at night (b) H0: p = 0.85 Ha: p ? 0.85 2. As part of its 2010 census marketing campaign, the U.S. Census Bureau advertised “10 questions, 10 minutes—that’s all it takes.” On the census form itself, we read, “The U.S. Census Bureau estimates that, for the average household, this form will take about 10 minutes to complete, including the time for reviewing the instructions and answers.” We suspect that the actual time it takes to complete the form may be longer than advertised. (a) µ = true mean amount of time that it takes to complete the census form. (b) H0: µ = 10 Ha: µ > 10 381000064452500**Common Exam Errors: Don’t use statistics in the hypotheses. Statistics are found after you gather data, which you are doing to evaluate the claim you make in the hypotheses. Also, make sure you are using the symbols p or µ rather than p or x, because we are trying to make inferences about the population parameter, not the sample statistic. Good: H0: µ = 10 Bad: H0: x = 10 Ha: µ > 10 Ha: x > 10 The P-value is the probability of an event happening assuming H0 is true. (We find this in the DO step) 5312410127635How sig. tests are related to confidence intervals 020000How sig. tests are related to confidence intervals 4019550-44454962525-4445 p < 0.01 strong evidence to reject H0 > 99% 0.01 < p < 0.05 enough evidence to reject H0 95% - 99% 0.05 < p < 0.10 possible evidence to reject H0 90% - 95% P > 0.10 not enough evidence to reject H0 < 90% Example, p. 544 Healthy Bones Calcium is a vital nutrient for healthy bones and teeth. The National Institute of Health (NIH) recommends a calcium intake of 1300 mg per day for teenagers. The NIH is concerned that teenagers aren’t getting enough calcium. Is this true? Researchers want to perform a test of H0: µ = 1300 Ha: µ < 1300 where µ is the true mean daily calcium intake in the population of teenagers. They ask a random sample of 20 teens to record their food and drink consumption for 1 day. The researchers then compute the calcium intake for each student. Data analysis reveals that x = 1198 mg and sx = 411 mg. After checking that conditions were met, researchers performed a significance test and obtained a P-value of 0.1404. (a) Explain what it would mean for the null hypothesis to be true in this setting. In this setting, H0: µ = 1300 says that the mean daily calcium intake in the population of teenagers is 1300 mg. If H0 is true, then teenagers are getting enough calcium, on average. (b) Interpret the P-value in context. Assuming that the mean daily calcium intake in the teen population is 1300 mg, there is a 0.1404 probability of getting a sample mean of 1198 mg or less just by chance in a random sample of 20 teens. **Drawing conclusions based on the P-value: Example, p. 545 Healthy Bones The large P-value of 0.1404 gives weak evidence against H0. We therefore fail to reject H0. Researchers do not have convincing evidence that teens are getting less than 1300 mg. Free Throws For Hypotheses: H0: p = 0.80 and Ha: p < 0.80 where p is the proportion of free throws a basketball player makes, a P-value of 0.0075 is calculated. The small P-value of 0.0075 gives strong evidence against H0. We therefore reject H0. We have convincing evidence that the player makes fewer than 80% of his free throws. **Sometimes, a significance level ? is set. For instance, ? = 0.05 or ? = 0.01. This is the threshold at which we decide if we reject or fail to reject H0. Statistically Significant at level ? If the P-value is smaller than alpha, we say that the results of a study are statistically significant at level ? In this case, we reject the null hypothesis H0, and conclude that there is convincing evidence in favor of the alternative hypothesis Ha. Example, p. 546 Better batteries A company has developed a new deluxe AAA battery that is supposed to last longer than its regular AAA battery. However, these new batteries are more expensive to produce, so the company would like to be convinced that they really do last longer. Based on years of experience, the company knows that its regular AAA batteries last for 30 hours of continuous use, on average. The company selects an SRS of 15 new batteries and uses them continuously until they are completely drained. The sample mean lifetime is x= 33.9 hours. A significance test is performed using the hypotheses: H0: µ = 30 hours Ha: µ > 30 hours Where µ is the true mean lifetime of the new deluxe AAA batteries. The resulting P-value is 0.0729. What conclusion would you make for each of the following significance levels? Justify your answer. (a) ? = 0.10 Because the P-value of 0.0729 is less than ? = 0.10, we reject H0. We have convincing evidence that the company’s deluxe AAA batteries last longer than 30 hours, on average. (b) ? = 0.05 Because the P-value of 0.0729 is greater than ? = 0.05, we fail to reject H0. We do not have convincing evidence that the company’s deluxe AAA batteries last longer than 30 hours, on average. You do have the option of choosing a significance level if one is not stated. If you do this, make sure to do it before you begin the problem. ***EXAM TIP: Conclusion to a significant test should include: 1) an explicit comparison of the P-value to a stated significance level, 2) a decision about the null hypothesis: reject or fail to reject H0, 3) a statement in the context of the problem about whether or not there is convincing evidence for Ha. Homework: p. 551 #2, 3, 8, 10, 25, 27, 28 Due: Monday