Question

2 pts We have a sample data as; 22.2 24.8 24.7 26.5 20.9 23.8 26 25.6 27 23.9 Find the p-value if sigma-1.5 or not at alfa-0.05. If true sigma is 0.75, what is the probability of making a type ll error? HTML Editor UA T Paragraph

Answer 1

Let us first consider the problem of testing the null hypothesis Ho that the population variance σ2 equals a specified value σ, against one of the usual alter- natives σ2 < σ1, σ2 > ơi , or σ2 σ . The appropriate statistic on which to base our decision is the chi-squared statistic of Theorem 8.4, which was used in Chapter 9 to construct a confidence interval for σ2. Therefore, if we assume that the distribution of the population being sampled is normal, the chi-squared value for testing σ-O is given by 2-(n-1)82 2 where n is the sample size, s2 is the sample variance, and σ is the value of σ2 given by the null hypothesis. If Ho is true, x2 is a value of the chi-squared distribution with u n-1 degrees of freedom. Hence, for a two-tailed test at the a-level of significance, the critical region 1S sided alternative σ2 < σ1, the critical region is χ2 < χ1-α, and for the one-sided alternative σ2 > σ χ2 < χ2-a/2 or χ2 > χ2/2. For the one- 2 0, the critical region is X>

Method

Null hypothesis         σ = 1.5
Alternative hypothesis σ ≠ 1.5

The chi-square method is only for the normal distribution.

                          Test
Variable Method      Statistic DF P-Value
C1        Chi-Square      14.63   9   0.203

p-value = 0.203

b)

Power and Sample Size

Test for One Standard Deviation

Testing StDev = null (versus ≠ null)
Calculating power for (StDev / null) = ratio
α = 0.05

       Sample
Ratio    Size     Power
0.5      10 0.710443

type ii error = - power = 1 - 0.710443 = 0.289557