Compare tests by Type II error probability

A question is this type if and only if it asks the student to compare two hypothesis tests (differing in sample size or significance level) and determine which has a smaller or larger probability of Type II error, with or without calculation.

2 questions · Standard +0.3

2.05e Hypothesis test for normal mean: known variance
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OCR S2 2006 June Q7
18 marks Standard +0.3
7 Three independent researchers, \(A , B\) and \(C\), carry out significance tests on the power consumption of a manufacturer's domestic heaters. The power consumption, \(X\) watts, is a normally distributed random variable with mean \(\mu\) and standard deviation 60. Each researcher tests the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 4000\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu > 4000\). Researcher \(A\) uses a sample of size 50 and a significance level of \(5 \%\).
  1. Find the critical region for this test, giving your answer correct to 4 significant figures. In fact the value of \(\mu\) is 4020 .
  2. Calculate the probability that Researcher \(A\) makes a Type II error.
  3. Researcher \(B\) uses a sample bigger than 50 and a significance level of \(5 \%\). Explain whether the probability that Researcher \(B\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
  4. Researcher \(C\) uses a sample of size 50 and a significance level bigger than \(5 \%\). Explain whether the probability that Researcher \(C\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
  5. State with a reason whether it is necessary to use the Central Limit Theorem at any point in this question.
OCR S2 2007 June Q8
13 marks Standard +0.3
8 A random variable \(Y\) is normally distributed with mean \(\mu\) and variance 12.25. Two statisticians carry out significance tests of the hypotheses \(\mathrm { H } _ { 0 } : \mu = 63.0 , \mathrm { H } _ { 1 } : \mu > 63.0\).
  1. Statistician \(A\) uses the mean \(\bar { Y }\) of a sample of size 23, and the critical region for his test is \(\bar { Y } > 64.20\). Find the significance level for \(A\) 's test.
  2. Statistician \(B\) uses the mean of a sample of size 50 and a significance level of \(5 \%\).
    1. Find the critical region for \(B\) 's test.
    2. Given that \(\mu = 65.0\), find the probability that \(B\) 's test results in a Type II error.
    3. Given that, when \(\mu = 65.0\), the probability that \(A\) 's test results in a Type II error is 0.1365 , state with a reason which test is better.