Chapter 27 & 29. One-tailed vs. Two-tailed Tests. Examples. Examples. Chapter 27 & 29. One-tailed vs. Two-tailed Tests. Examples.

Transcription

1 Interpreting the P-value Part VIII of Signicance More for Averages Chapter 29 A Closer Look at of Signicance Example 2 from earlier lecture A senator introduces bill that simplies the tax code. He claims that the bill is 'revenue neutral'. This means that the total tax revenues will stay the same. To check the claim, the Treasury Department uses a computer le of 100,000 representative tax returns. For a simple random sample of 100 returns, it calculates change = tax under new rules tax under old rules The sample average comes out as a change of -$219, with an SD of $725. We set up the following test ˆ Null hypothesis: the average of the box equals Interpreting the P-value two-tailed tests We have the following test: In this example, the two hypotheses do not cover all possible outcomes. ˆ Null hypothesis: the average of the box equals ˆ Null hypothesis: the average of the box equals If we cannot reject the null hypothesis, we don't know anything for sure (but we often make an educated guess). The truth could, in theory, also be that the true average of the box is larger than. If we reject the null hypothesis, the data is telling us that the statement in the null hypothesis is wrong (based on the data). That means the opposite, the alternative hypothesis, must be true. Remember, however, that the P-value is the chance of getting a test statistic as extreme as or more extreme than the one we saw, given that the null hypothesis is true. If we reject the null hypothesis because it is too unlikely that the true average is when the observed value is negative, we would also reject every value greater than.

2 Instead of two-tailed tests Test 1: Null hypothesis: average of the box equals Alternative hypothesis: average of the box is less than we could also test Test 2: We want to test whether a coin is fair. Does it land heads with probability 50%? We toss the coin 100 times and get 61 heads. The model consists of 100 draws from the box 0?? 1?? 0 = tails, 1 = heads The null hypothesis says that the fraction of 1's in the box is 1/2. The test statistics is Null hypothesis: average of the box equals Alternative hypothesis: average of the box is not equal to z= observed - expected SE = = like Test 1 are called one-tailed tests, while tests like Test 2 are called two-tailed tests. tests are covered in Section 2 of Chapter 29. Alternative hypothesis nr. 1: the coin is biased towards heads; the fraction of 1's in the box is larger than 1/2. Alternative hypothesis nr. 2: the fraction of 1's in the box di ers from 1/2, and may be bigger or smaller. The P -value is the area to the right of 2.2 under the normal curve: This is a one-tailed test. Now the P -value is This is a two-tailed test.

3 two-tailed tests two samples is about comparing two samples. Box A Box B Average = 110 SD = 60 Average = 90 SD = 40 Four hundred draws are made at random with replacement from box A, and independently 100 draws are made at random with replacement from box B. We are interested in the di erence Box A? +? + +? 400 Box B? +? + +? 100 Example 2 (a) One hundred draws are made at random from box X. The average of the draws is 51.8, and their SD is 9. The null hypothesis says that the average of the box equals 50, while the alternative hypothesis says that the average of the box di ers from 50. Is a one-tailed or a two-tailed z -test more appropriate? (b) One hundred draws are made at random from box Y. The average of the draws is 51.8, and their SD is 9. The null hypothesis says that the average of the box equals 50, while the alternative hypothesis says that the average of the box is bigger than 50. Is a one-tailed or a two-tailed z -test more appropriate? two samples We need the expected value and the standard error for the di erence between the two sample averages. First, we nd the EV and the SE for each box: Average of 400 draws from box A = 110 ± 3 or so Average of 100 draws from box B = 90 ± 4 or so The expected value for the di erence is just = 20. The SE is more complicated Standard error for the di erence of two independent quantities is q SEa2 + SEb2, where a denotes the rst quantity and b the second quantity.

4 two samples Example 3: Cholesterol Here, the standard error for the dierence is thus given by SE for dierence = = 5, as the two samples are independent. We can use this method to compare the averages and the percentages for two independent samples, but we CANNOT compare the sum or the count of two samples. Note: Generally, the formulas give the wrong answer when applied to two dependent samples. There is an exception: we can use them to compare the treatment and the control group in a randomized controlled experiment even though the groups are dependent. A randomized controlled double-blind experiment was performed to demonstrate the ecacy of a drug called 'cholestyramine' in reducing blood cholesterol levels and preventing heart attacks. There were 3,806 subjects, who were all middle-aged men at high risk of heart attack; 1,906 were chosen at random for the treatment group and 1,900 were assigned to the control group. Subjects were followd for 7 years. There were 155 heart attacks in the treatment group and 187 in the control group: 8.1% 9.8%. Is there a dierence? (a) There is one ticket in the box for each... (b) The ticket is marked... for the men and... for the women. (c) The number of tickets in the box is... and the number of draws is... (d) The null hypothesis says that the sample is like made at random from the box. (e) The percentage of 1's in the box is... (f) The observed number of men is... (g) The expected number of men is...

5 (h) If the null hypothesis is right, the number of men in the sample is like the... of the draws from the box. (i) The SE for the number of men is... (j) z =... and P =... (k) Was Alpert's sampling procedure like taking a simple random sample? Explain.