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AP Statistics Chapter 9 Scavenger Hunt Review Power your goal and objective is to be able to REJECT the null hypothesis ... in the case of a simple experiment perhaps ... to show that the new method works better than the old method. "power" is being able to REJECT the null when the treatment really makes a difference. So, we want to design the study and set the study conditions and implement the study in such a way as to give yourself a good chance of being able to REJECT the null where there really is an effect. I think this is the best place to focus on that 2 by 2 table of: Reality .... Null is true ... Null is false What you do ... Retain the null ... Reject the null What we hope to do in the study is to REJECT the null when the reality is that the null is false. What impacts the power of your test? 1) The effect size: the greater the improvement, the more likely you are to notice. If your new treatment raises the cure rate to 90%, that should be pretty obvious. But if it only improves relief to 65%, you're probably not going to spot that. Larger effect = more power. (And less chance of overlooking the improvement = Type II error.) 2) The sample size: the more people who try your treatment, the better your chances of detecting an improvement. Testing the new drug on 1000 people makes you more likely to spot an effect than if you had only 20 people try it. Larger sample = more power. 3) Your convinceability (if that's a word): the weaker the evidence you are willing to accept, the more easily you'll notice an improvement. If you are willing to believe marginal evidence, you'll spot a higher cure rate more easily than if you insist on compelling proof. Easier to convince = more power. (But it also makes you more vulnerable to thinking the new treatment is better when it really isn't = Type I error.) Increasing Power • (if you increase alpha, increase sample size, decrease sigma is in our book, but I've always wondered how you can change sigma, and having an alternative hypothesis farther away from the null hypothesis.) Directions • Print slides and mount on the walls in your room, but in a random order. • Assign students to start on a different problem. • Answer to previous problem is at top of the next slide. 0.0423 • State the null hypothesis and alternative hypothesis • During the winter months, the temperatures at the Colorado cabin owned by the Starnes family can stay well below freezing for weeks at a time. To prevent the pipes from freezing, Mrs. Starnes sets the thermostat at 50o F. The manufacturer claims that the thermostat allows variation in home temperature of 3o F. Mrs. Starnes suspects that the manufacturer is overstating how ell the thermostat works. Null: Sigma=3 Alternative: Sigma>3 • A Gallup Poll report on a national survery of 1028 teenagers revealed that 72% of teens said they rarely or never argue with their friends. Yvonne wonders whether this national result would be true in her large high school, so she surveys a random sample of 150 students at her school. She finds that 96 students seldom or rarely argue with friends. What is the test statistic and p-value? -2.18, .0292 • A state’s DMV claims that 60% of teens pass their driving test on the first attempt. An investigative reporter examines an SRS of the DMV records for 125 teens; 86 of them passed the test on their first try. Is this good evidence that the DMV’s claim is incorrect? Do we accept or reject the null at a .05 level? Reject • How well materials conduct heat matters when designing houses, for example. Conductivity is measured in terms of watts of heat power transmitted per square meter of surface per degree Celsius of temperature difference on the two sides of the material. In these units, glass has conductivity about 1. The NIST provides exact data on properties of materials. Here are measurements of the heat conductivity of 11 randomly selected pieces of a particular type of glass: 1.11, 1.07, 1.11, 1.07, 1.12, 1.08, 1.08, 1.18, 1.18, 1.18,1.12. Is there convincing evidence that the conductivity of this type of glass is greater than 1. What is the p-value? Do we accept or reject the null? 0, Reject • For students with special preparation, SAT Math scores in recent years have varied Normally with mean of 518. One hundred students go through a rigorous training program designed to raise their SAT Math scores by improving their mathematics skills. Carry out a test for the null=518, the alternative being greater than 518. If the sample has a mean of 536.7 and a std. dev. Of 114, is the result significant at the 5% level? No • Radon is a colorless, odorless gas that is naturally released by rocks and soils and may concentrate in tightly closed houses. Because radon is slightly radioactive, there is some concern that it may be a health hazard. Radon detectors are sold to homeowners worried about this risk, but the detectors may be inaccurate. University researches placed a random sample of 11 detectors in a chamber where they were exposed to 105 picocuries per liter of radon over 3 days. A graph of the radon readings from the 11 detectors show no strong skewness or outliers. The minitab output shows the results of a one-sample t interval. Is there significant evidence at the 10% level that the mean reading differs from the true value 105? n 11, mean 104.82, stdev 9.54, SEmean 2.88,90%CI : 99.61,110.03 Stop Here • Stop Here 0.8351 • A promoter knows that 23% of males enjoy watching boxing matches. In a random sample of 125 men, what is the distribution, mean and standard deviation of the sample? Approx normal with mean = 0.23, std dev = 0.38 • A promoter knows that only 12% of females enjoy watching boxing matches. In a random sample of 125 women, what is the probability that more than 10% of the females enjoy watching boxing matches? 0.7548 • The average number of missed school days for students in public schools is 8.5 with a standard deviation of 4.1. In a sample of 200 public school students, what is the probability that the average number of days missed is less than 8 days? Answers • Adult women & time for themselves: – Approx normal, μ= .47, σ = .016 – Np = 1025*.47 = 481.75 – Nq = 1025 *.53 = 543.25 • Internet access fees – Approx normal with μ= 28, σ = 0.447 – P(x-bar > 29) = 0.0126 Answers • Mail order company – Approx normal with μ= .9, σ = .03 – Np = .9 * 100 = 90, nq = .1 * 100 = 10 – P(p-hat ≤ .86) = 0.0912 • Age of cars – Approx normal, mean = 4, std dev = 0.11 Answers • Car Dealership service – Approx normal with mean = 0.61, std dev = 0.049 – Np = 61, nq = 39 • College teacher incomes – P(x-bar ≥ 40,000) = 0.228 with std dev = 2500 Answers • Paper strength – Std dev = .36 – P(24.5 ≤ x – bar ≤ 25.5) = .8351 • Males & boxing matches – Approx normal with mean = .23, std dev = 0.38 – Np = .23 * 125 = 28.75 – Nq = .77 * 125 = 96.25 Answers • Females & boxing matches – Approx normal with mean = .12, std dev = .029 – Np = 125 * .12 = 15 – Nq = 125 *.88 = 110 – P(p=hat ≥ 0.1) = .7548 • Missed school days – Std dev = .29 – P(x-bar < 8) = .0423