Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Bootstrapping (statistics) wikipedia , lookup
Degrees of freedom (statistics) wikipedia , lookup
Psychometrics wikipedia , lookup
Taylor's law wikipedia , lookup
Foundations of statistics wikipedia , lookup
Analysis of variance wikipedia , lookup
Statistical hypothesis testing wikipedia , lookup
Misuse of statistics wikipedia , lookup
Lecture 14: Hypothesis testing, continued Announcements: Reading for this week’s subject: pp. 61 - 79 in Vasilj. Hypothesis testing: example Hypothesis testing: example We have a new variety of tomato. Hypothesis testing: example We have a new variety of tomato. Does it have the same resistance to fusarium wilt as the old variety? Hypothesis testing: example We have a new variety of tomato. Does it have the same resistance to fusarium wilt as the old variety? To answer this question, we do an experiment. Hypothesis testing: example We have a new variety of tomato. Does it have the same resistance to fusarium wilt as the old variety? To answer this question, we do an experiment. We expose 100 new plants to fusarium, and found that the mean growth rate was 35 with standard deviation 10. Hypothesis testing: example We have a new variety of tomato. Does it have the same resistance to fusarium wilt as the old variety? To answer this question, we do an experiment. We expose 100 new plants to fusarium, and found that the mean growth rate was 35 with standard deviation 10. We know from previous experience that the mean growth rate of the old genotype is 33. Hypothesis testing: procedure 1 We ask a yes/no question about a population. Hypothesis testing: procedure 1 2 We ask a yes/no question about a population. We answer the question yes, and answer the question no, using symbols for the population means. Hypothesis testing: procedure 1 2 3 We ask a yes/no question about a population. We answer the question yes, and answer the question no, using symbols for the population means. We label one answer the null hypothesis and the other answer the alternative hypothesis. Hypothesis testing: procedure 1 2 3 4 We ask a yes/no question about a population. We answer the question yes, and answer the question no, using symbols for the population means. We label one answer the null hypothesis and the other answer the alternative hypothesis. We decide the criterion for rejecting the null hypothesis. Hypothesis testing: procedure 1 2 3 4 5 We ask a yes/no question about a population. We answer the question yes, and answer the question no, using symbols for the population means. We label one answer the null hypothesis and the other answer the alternative hypothesis. We decide the criterion for rejecting the null hypothesis. The test is one of: two-tailed, right-tailed, or left-tailed. Hypothesis testing: procedure 1 2 3 4 5 6 We ask a yes/no question about a population. We answer the question yes, and answer the question no, using symbols for the population means. We label one answer the null hypothesis and the other answer the alternative hypothesis. We decide the criterion for rejecting the null hypothesis. The test is one of: two-tailed, right-tailed, or left-tailed. We take a sample, and calculate our test statistic (Z or t for now) Hypothesis testing: procedure 1 2 3 4 5 6 7 We ask a yes/no question about a population. We answer the question yes, and answer the question no, using symbols for the population means. We label one answer the null hypothesis and the other answer the alternative hypothesis. We decide the criterion for rejecting the null hypothesis. The test is one of: two-tailed, right-tailed, or left-tailed. We take a sample, and calculate our test statistic (Z or t for now) We find if the observed test statistic is in the rejection region (critical region or tail) of the distribution. Hypothesis testing: procedure 1 2 3 4 5 6 7 8 We ask a yes/no question about a population. We answer the question yes, and answer the question no, using symbols for the population means. We label one answer the null hypothesis and the other answer the alternative hypothesis. We decide the criterion for rejecting the null hypothesis. The test is one of: two-tailed, right-tailed, or left-tailed. We take a sample, and calculate our test statistic (Z or t for now) We find if the observed test statistic is in the rejection region (critical region or tail) of the distribution. If the statistic is in the rejection region, we reject the null hypothesis and accept the alternative hypothesis. Hypothesis testing: procedure 1 2 3 4 5 6 7 8 9 We ask a yes/no question about a population. We answer the question yes, and answer the question no, using symbols for the population means. We label one answer the null hypothesis and the other answer the alternative hypothesis. We decide the criterion for rejecting the null hypothesis. The test is one of: two-tailed, right-tailed, or left-tailed. We take a sample, and calculate our test statistic (Z or t for now) We find if the observed test statistic is in the rejection region (critical region or tail) of the distribution. If the statistic is in the rejection region, we reject the null hypothesis and accept the alternative hypothesis. If the statistic is not in the rejection region, we retain the null hypothesis, and do not accept the alternative hypothesis. Rules If the alternative hypothesis contains the 6= sign, the test is two-tailed. Rules If the alternative hypothesis contains the 6= sign, the test is two-tailed. If the alternative hypothesis contains the < sign, the test is left-tailed. Rules If the alternative hypothesis contains the 6= sign, the test is two-tailed. If the alternative hypothesis contains the < sign, the test is left-tailed. If the alternative hypothesis contains the > sign, the test is right-tailed. Hypothesis testing: example We have a new variety of tomato. Does it have the same resistance to fusarium wilt as the old variety? To answer this question, we do an experiment. We expose 100 new plants to fusarium, and found that the mean growth rate was 35 with standard deviation 10. We know from previous experience that the mean growth rate of the old genotype is 33. Hypothesis testing: example Hypothesis testing: example We have a new variety of tomato. Does it have lower resistance to fusarium wilt than the old variety? To answer this question, we do an experiment. We expose 100 new plants to fusarium, and found that the mean growth rate was 35 with standard deviation 10. We know from previous experience that the mean growth rate of the old genotype is 33. Hypothesis testing: example Hypothesis testing: example We have a new variety of tomato. Does it have higher resistance to fusarium wilt than the old variety? To answer this question, we do an experiment. We expose 100 new plants to fusarium, and found that the mean growth rate was 35 with standard deviation 10. We know from previous experience that the mean growth rate of the old genotype is 33. Hypothesis testing: example Some terminology If we reject the null hypothesis, then we conclude our data are significant, or significantly different from the null. Some terminology If we reject the null hypothesis, then we conclude our data are significant, or significantly different from the null. The probability of the test is the probability of getting the observed test result, or more extreme, than expected by chance, if the null hypothesis were true. Some terminology If we reject the null hypothesis, then we conclude our data are significant, or significantly different from the null. The probability of the test is the probability of getting the observed test result, or more extreme, than expected by chance, if the null hypothesis were true. In a two-tailed test, the probability is twice the tail probability. In a one-tailed test, the probability is the one-tail probability. Some terminology If we reject the null hypothesis, then we conclude our data are significant, or significantly different from the null. The probability of the test is the probability of getting the observed test result, or more extreme, than expected by chance, if the null hypothesis were true. In a two-tailed test, the probability is twice the tail probability. In a one-tailed test, the probability is the one-tail probability. This probability is also called the significance or the significance level of the test. Some more terminology The least significant difference: Some more terminology The least significant difference: Some more terminology The least significant difference: t= observed − expected sobserved Some more terminology The least significant difference: t= observed − expected sobserved tcrit = observed − expected sobserved Some more terminology The least significant difference: t= observed − expected sobserved tcrit = observed − expected sobserved LSD = Some more terminology The least significant difference: t= observed − expected sobserved tcrit = observed − expected sobserved LSD = tcrit Some more terminology The least significant difference: t= observed − expected sobserved tcrit = observed − expected sobserved LSD = tcrit sobserved Hypothesis testing: example We have a new variety of tomato. Does it have the same resistance to fusarium wilt as the old variety? To answer this question, we do an experiment. We expose 100 new plants to fusarium, and found that the mean growth rate was 35 with standard deviation 10. We know from previous experience that the mean growth rate of the old genotype is 33. Hypothesis testing: example Hypothesis testing: example Hypothesis testing: example We have a new variety of tomato. Does it have the same resistance to fusarium wilt as the old variety? To answer this question, we do an experiment. We expose 100 new plants and 100 old plants to fusarium, and found that the mean growth rate was 35 with standard deviation 10 for the new plants, and mean 33 and standard deviation 8 for the old plants. Hypothesis testing: example Unequal variance t-test Calculate separate variances for the two samples: Unequal variance t-test Calculate separate variances for the two samples: t= Unequal variance t-test Calculate separate variances for the two samples: t= x1 − x2 Unequal variance t-test Calculate separate variances for the two samples: t=q x1 − x2 s12 /n1 + s22 /n2 Unequal variance t-test Calculate separate variances for the two samples: t=q x1 − x2 s12 /n1 + s22 /n2 What is the degrees of freedom? Unequal variance t-test Calculate separate variances for the two samples: t=q x1 − x2 s12 /n1 + s22 /n2 What is the degrees of freedom? Either use the smallest of n1 − 1 and n2 − 1, Unequal variance t-test Calculate separate variances for the two samples: t=q x1 − x2 s12 /n1 + s22 /n2 What is the degrees of freedom? Either use the smallest of n1 − 1 and n2 − 1, or use R to calculate the following degrees of freedom: Unequal variance t-test Calculate separate variances for the two samples: t=q x1 − x2 s12 /n1 + s22 /n2 What is the degrees of freedom? Either use the smallest of n1 − 1 and n2 − 1, or use R to calculate the following degrees of freedom: d.f. = Unequal variance t-test Calculate separate variances for the two samples: t=q x1 − x2 s12 /n1 + s22 /n2 What is the degrees of freedom? Either use the smallest of n1 − 1 and n2 − 1, or use R to calculate the following degrees of freedom: d.f. = (s12 /n1 + s22 /n2 )2 Unequal variance t-test Calculate separate variances for the two samples: t=q x1 − x2 s12 /n1 + s22 /n2 What is the degrees of freedom? Either use the smallest of n1 − 1 and n2 − 1, or use R to calculate the following degrees of freedom: d.f. = (s12 /n1 + s22 /n2 )2 . (s12 /n1 )2 /(n1 − 1) + (s22 /n2 )2 /(n2 − 1) Hypothesis testing: example Equal variance t-test: use the pooled variance to calculate the sample standard deviation Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = (n1 − 1)si2 + Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = (n1 − 1)si2 + (n2 − 1)s22 Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = s= (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = s= p (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 s s= p pooled variance = Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 s s= p pooled variance = (n1 − 1)si2 + Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 s s= p pooled variance = (n1 − 1)si2 + (n2 − 1)s22 Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 s s= p pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 s s= p pooled variance = t= (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 s s= p pooled variance = t= x1 − x2 (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 s s= p pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 x1 − x2 t=p = 2 s /n1 + s 2 /n2 Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 s s= p pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 x1 − x2 t=p = 2 s /n1 + s 2 /n2 x1 − x2 Equal variance t-test: use the pooled variance to calculate the sample standard deviation pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 s s= p pooled variance = (n1 − 1)si2 + (n2 − 1)s22 n1 + n2 − 2 x1 − x2 x1 − x2 t=p = p 2 2 s 1/n1 + 1/n2 s /n1 + s /n2 Hypothesis testing: example We have a new variety of tomato. Does it have the same resistance to fusarium wilt as the old variety? To answer this question, we do an experiment. We expose 100 new plants and 100 old plants to fusarium, and found that the mean growth rate was 35 with standard deviation 10 for the new plants, and mean 33 and standard deviation 8 for the old plants. Hypothesis testing: example Hypothesis testing: one-sample test of proportion example We have a new variety of tomato. Does it have the same resistance to fusarium wilt as the old variety? To answer this question, we do an experiment. We expose 100 new plants to fusarium, and found that 72 are resistant. We know in advance that 65% of the old plants are resistant. Hypothesis testing: example Hypothesis testing: two-sample test of proportions example We have a new variety of tomato. Does it have the same resistance to fusarium wilt as the old variety? To answer this question, we do an experiment. We expose 100 new plants and 100 old plants to fusarium, and found that 72 new are resistant, and 65 old are resistant. Hypothesis testing: example Two sample test of proportions Given x1 successes in a sample of n1 , and x2 successes in a sample of n3 . Therefore pˆ1 = x1 /n1 , and pˆ2 = x2 /n2 . +x2 We define p = nx11 +n , and q = 1 − p. 2 Then our test statistic is: pˆ1 − pˆ2 Z=q pq pq n1 + n2