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Unformatted text preview: Chapter 10a: One-Sample Tests of Hypotheses 115 CHAPTER 10a — ONE-SAMPLE TESTS OF HYPOTHESES In Chapter 9a, we illustrated how to construct interval estimates for an unknown population mean based on information from a random sample. However, the purpose of statistical inference is not merely to estimate parameter values. Rather, we often wish to make judgments about a parameter value and base some action on this judgment. Essentially, we will be evaluating claims, termed hypotheses , using statistics calculated from random samples. Motivating Example: A Coin Story Suppose your instructor hands you a coin, claiming that it is balanced ( i.e. , P[H] = 0.5). However, you know that this instructor likes to play tricks on people, and you suspect the coin might actually be imbalanced. Instructor’s claim b Balanced coin b P[H] = .5 b Null hypothesis Your suspicion b Imbalanced coin b P[H] ≠ .5 b Alternative hypothesis You would like to find a way to test whether or not the coin is actually fair. So, you flip the coin 20 times. Intuitively, if the coin shows heads about half of the time, you might tend to believe the instructor. However, if the coin lands heads much more than or much fewer than 10 times, your suspicion that the coin is imbalanced will grow. Formally, you begin by assuming that the instructor’s claim (the null hypothesis) is true . In other words, you initially give the instructor the benefit of the doubt. Then, you’ll try to determine how “unusual” your experimental data are in light of this assumption. We can do this by thinking about all of the 20 2 = 1,048,576 possible sequences of 20 coin flips. We can “rank” all of these possible sequences beginning with the 184,756 (10 H, 10 T) sequences, which are most favorable to the instructor’s claim, and ending with the (0 H, 20 T) and (20 H, 0 T) sequences, which are least favorable to the instructor’s claim. Since a fair coin implies that each of the sequences is equally likely, we can determine the probability that an outcome “as extreme (unfavorable to the instructor’s claim) or more extreme” than our observed sequence would occur. Chapter 10a: One-Sample Tests of Hypotheses 116 # sequences As extreme or more extreme most favorable 10 H & 10 T 184,756 1,048,576 100.0000 % ↑ (9 H, 11 T) or (11 H, 9 T) 335,920 863,820 82.3803 % | (8 H, 12 T) or (12 H, 8 T) 251,940 527,900 50.3445 % | (7 H, 13 T) or (13 H, 7 T) 155,040 275,960 26.3176 % | (6 H, 14 T) or (14 H, 6 T) 77,520 120,920 11.5318 % | (5 H, 15 T) or (15 H, 5 T) 31,008 43,400 4.1389 % | (4 H, 16 T) or (16 H, 4 T) 9,690 12,392 1.1818 % | (3 H, 17 T) or (17 H, 3 T) 2,280 2,702 0.2577 % | (2 H, 18 T) or (18 H, 2 T) 380 422 0.0402 % ↓ (1 H, 19 T) or (19 H, 1 T) 40 42 0.0040 % least favorable (0 H, 20 T) or (20 H, 0 T) 2 2 0.0002 % • Suppose we encounter a sequence of 10 heads and 10 tails. Even though we observed one of the sequences most favorable to the fair-coin claim, this does not prove (“for sure”) that the coin is fair. sure”) that the coin is fair....
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- Spring '08
- Statistics, Null hypothesis, Statistical hypothesis testing, rejection region, Hypotheses Terminology of Hypothesis Testing, one-sample tests