4. True or False? (No proofs required) of the two groups to be equal interval ar

Question

4. True or False? (No proofs required) of the two groups to be equal interval are equal in large samples. bility of a test .999. alternative that it is > 10, the type I error probability and the power of any (a) The two sample t confidence interval necessarily needs the variances (b) The length of the (e) B (d) For testing that the mean of a normal distribution is 10 against the Rao score interval and the length of the Wald y using a large enough sample, we can make the type I error proba- test are both functions of the unknown mean. (e) It is impossible to make both a type I and a type II error in a particular testing problem.

Explanation / Answer

(4a) FALSE. If the variances are unequal, we perform a test called Welch t-test where we find out the combined variance of the sample assuming unequal variance using the individual variances's unbiased estimators.. The test remains a 2 sample t-test.

(b) TRUE. Both the Rao's score testa and Wald score test are assymptotically similar test because both are compared against a chi-square distribution of different degrees of freedom but as n increases the df also become almost equal nad hence the intervals become almost equal.

(c) TRUE. The result is intutitive. The larger the sample size, we are more sure about the outcome of the test. Hence the type ! error reduces and the power of the test increases.

(d) FALSE. The type 1 error can be calculated as the max of type 1 errors among al values of mean. So type ! error can actually be calculated using mean value of 10, because it will give the max Type ! error also called the size of the test. However power of the test do depend on the actual alternative value of the mean.

(e) TRUE. In a testing problem we either accept the Null hypotheses (then there is chance of type 2 error, no chance of type 1 error) or we reject the Null Hyp (there is chancce of type 1 error and no chance of type 2 error)

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