Hypothesis Testing One Sample Tests

Transcription

1 STATISTICS Lecture no. 13 Department of Econometrics FEM UO Brno office 69a, tel

2 Tests on Mean of a Normal distribution Tests on Variance of a Normal distribution Test of the Parameter µ of a Normal Distribution Let X 1, X 2,..., X n be a random sample from a normal distribution N(µ, σ 2 ). A statistic T = X µ n S has a Student distribution with ν = n 1 degrees of freedom. We use this statistic for testing of the parameter µ.

3 Tests on Mean of a Normal distribution Tests on Variance of a Normal distribution Test of the Parameter µ of a Normal Distribution Let x 1, x 2,..., x n be values of a random sample (measured data), x denote an arithmetic mean and s a sample standard deviation. We test the hypothesis that the parameter µ is equal to a constant µ 0 : H : µ = µ 0, the test statistic t = x µ 0 n, s has under the null hypothesis H a Student t-distribution with ν = n 1 degrees of freedom.

4 Tests on Mean of a Normal distribution Tests on Variance of a Normal distribution Test of the Parameter µ of a Normal Distribution According to the alternative hypothesis we construct following regions of rejection: alternative hypothesis A : µ > µ 0 rejection region W α = {t, t t 1 α (ν)} A : µ < µ 0 W α = {t, t t 1 α (ν)} { } A : µ µ 0 W α = t, t t 1 α (ν) 2 where t 1 α, t 1 α 2 are quantiles of the Student distribution.

5 Tests on Mean of a Normal distribution Tests on Variance of a Normal distribution Test of the Parameter σ 2 of a Normal distribution Let X 1, X 2,..., X n be a random sample from a normal distribution N(µ, σ 2 ). A statistic χ 2 (n 1)S 2 = σ 2 has a Pearson distribution with ν = n 1 degrees of freedom. We use this statistic for testing of the parameter σ 2.

6 Tests on Mean of a Normal distribution Tests on Variance of a Normal distribution Test of the Parameter σ 2 of a Normal distribution Let x 1, x 2,..., x n be values of a random sample (measured data), s 2 denote sample variance. We test the hypothesis that the parameter σ 2 is equal to a constant σ 2 0 : H : σ 2 = σ 2 0, the test statistic χ 2 = (n 1)s2 σ 2 0 has under the null hypothesis H a Pearson χ 2 -distribution with ν = n 1 degrees of freedom.

7 Tests on Mean of a Normal distribution Tests on Variance of a Normal distribution Test of the Parameter σ 2 of a Normal distribution According to an alternative hypothesis we construct following regions of rejection: alternative hypothesis rejection region A : σ 2 > σ0 2 W α = { χ 2, χ 2 χ 2 1 α (ν)} A : σ 2 < σ0 2 W α = { χ 2, χ 2 χ 2 α(ν) } { A : σ 2 σ0 2 W α = χ 2, χ 2 χ 2 α (ν) or χ 2 χ α 2 } (ν) where χ 2 1 α, χ2 1 α 2 are quantiles of the Pearson χ 2 -distribution.

8 Let X 1, X 2,..., X n be a random sample from any distribution with the mean µ. A statistic U = X µ n S has for large n approximately a normal distribution N(0, 1) see the central limit theorems. We use this statistic for testing of the parameter µ.

9 Let x 1, x 2,..., x n be values of a random sample (measured data), x denote an arithmetic mean and s a sample standard deviation. We test the hypothesis that the parameter µ is equal to a constant µ 0 : H : µ = µ 0, the test statistic u = x µ 0 n, s has under the null hypothesis H asymptotically a normal distribution N(0, 1).

10 According to an alternative hypothesis we construct following regions of rejection: alternative hypothesis rejection region A : µ > µ 0 W α = {u, u u 1 α } A : µ < µ 0 W α = {u, u u 1 α { } A : µ µ 0 W α = u, u u 1 α 2 where u 1 α, u 1 α 2 are quantiles of N(0, 1).

11 Suppose that a random sample of size n has been taken from a large (possibly infinite) population and that m observations in this sample belong to a class of interest. Then p = m n is a point estimator of the proportion of the population π that belongs to this class. A random variable U = ˆπ π π(1 π)/n has for n approximately a normal distribution N(0, 1) see central limit theorems. We use this statistic for testing of the population proportion.

12 Let ˆπ = m n be a point estimator of population proportion. We test the hypothesis that the parameter π is equal to a constant π 0 : H : π = π 0, a test statistic u = ˆπ π 0 π0 (1 π 0 )/n has under the null hypothesis H asymptotically a normal distribution N(0, 1).

13 According to an alternative hypothesis we construct following regions of rejection: alternative hypothesis rejection region A : π > π 0 W α = {u, u u 1 α } A : π < π 0 W α = {u, u u 1 α { } A : π π 0 W α = u, u u 1 α 2 where u 1 α, u 1 α 2 are quantiles of N(0, 1).