5. Let us again examine the original random sample of the canister filling example N(μ, σ²) i.e the model Y₁, . . ., Y25 ~ N(μ, σ²) H, where both μ and σ² are unknown parameters. You should now derive a test for the variance σ² (or equivalently for the standard de- viation σ). More precisely, let's look at a one-way test set-up in which the hypotheses are Ho: 20.202 and H₁: 0² > 0.202. = See section 5.6.4 of the lecture handout (page 53). Based on this, we conclude select t(y) (n - 1)s²/o as the test variable, where s² is the sample variance s² (y) = 1 Σi (Yi - y)² and σ is the variance corresponding the null hypothesis. Use the section 5.6.4 of the lecture handout to answer the following questions: n-1 (a) What values of the test statisics (large or small) are critical for Ho and support H₁? (hint: remember that s² is an estimate of σ² and H₁: σ² > 0.202. What do these tell together?) (b) What is the distribution of the random variable t(Y) corresponding to the test statistics when the null hypothesis Ho: σ² = 0.20² holds? = (c) Let's recall that based on the observed data, s = 0.22 and y 9.9. Using the definition of p-value (at least when large values are critical) justify why p-value of the test statistics t can be calculated with R by the following expression 1 - - pchisq (t, df = 24) where t is the value of the test variable t(y). Would the test reject the null hypothesis with the significance level a = 0.01? (hint: you should look at the help texts of R how the cumulative distribution function help (pchisq) behaves.)

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter13: Probability And Calculus

Section13.2: Expected Value And Variance Of Continuous Random Variables

Problem 10E