Statistics: finding a critical region

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In summary, a critical region in statistics refers to a specific range of values used to determine the rejection of a null hypothesis. It is determined by the significance level, or alpha level, which is the probability of making a type I error. The significance level is compared to the p-value, and if the p-value is less than or equal to the significance level, the null hypothesis is rejected and the critical region is established. If the test statistic falls within the critical region, it indicates that the observed data is unlikely to have occurred under the null hypothesis, leading to its rejection. The critical region can be changed by adjusting the significance level, but this also affects the risk of making a type I error.
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Aria1
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Let X1, X2, …, X10 be a random sample of size ten from the Normal(3, σ^2) distribution.
(a) Use the likelihood ratio test to derive a 5%-level critical region for testing H0 : σ^2 = 1
versus H1 : σ^2 ≠ 1.
(b)Suppose the following ten values from the Normal(3, σ^2) distribution are observed. Do we
reject the null hypothesis H0 : σ^2 = 1 at 5%-level? Please justify your answer.
2.82, 2.35, 3.74, 1.93, 4.51, 3.17, 1.41, 4.23, 4.77, 3.89
 
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Hey Aria1.

What is the definition of the LRT? (Hint: It involves the MLE likelihood and the likelihood of the null hypothesis).
 

FAQ: Statistics: finding a critical region

What is a critical region in statistics?

A critical region in statistics is a specific range of values that is used to determine the rejection of a null hypothesis. It is also referred to as the rejection region.

How is a critical region determined?

A critical region is determined by the significance level or alpha level, which is the probability of making a type I error. This value is compared to the p-value, and if the p-value is less than or equal to the significance level, the null hypothesis is rejected and the critical region is established.

What is the significance level and how does it relate to the critical region?

The significance level, also known as alpha level, is the probability of making a type I error, which is rejecting a true null hypothesis. The critical region is determined based on this value, as it represents the range of values where the null hypothesis will be rejected.

What happens if the test statistic falls within the critical region?

If the test statistic falls within the critical region, it means that the observed data is unlikely to have occurred under the assumption of the null hypothesis. This leads to the rejection of the null hypothesis and acceptance of the alternative hypothesis.

Can the critical region be changed?

Yes, the critical region can be changed by adjusting the significance level. A higher significance level will result in a larger critical region, making it easier to reject the null hypothesis. However, this also increases the risk of making a type I error.

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