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.)
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.)
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text: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.)
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