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Paper 2 – Type I and Type II Errors
Example
The random variable X follows a Poisson distribution with parameter � , representing the rate of occurrences of an event per hour .
A hypothesis test is conducted at a particular significance level to test whether � is less than 5 .
( a ) ( i ) Write down the null hypothesis of the test .
( ii ) Write down the alternative hypothesis of the test .
[ 2 ] The null hypothesis is rejected if it is observed that at most one occurrence of the event is observed in a particular hour .
( b ) Find the probability that a Type I error is made .
The actually value of � is 6 .
( c ) Find the probability that a Type II error is made .
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Another random variable Y follows a Binomial distribution with parameters n and q , representing the number of trials in an experiment and the probability of success of an event respectively .
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A hypothesis test is conducted at a 5 % significance level to test whether q is less than 0.25 . From a random sample of size 20 , there are 3 observed successes .
( d ) Write down the critical region for testing q , giving the answer in the form a �Y � b .
( e ) Find the p -value .
( f ) State the conclusion of the test with a reason .
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