3 . The random variable X follows a normal distribution with parameter � , where � represents the mean value of a population . The population variance is 5 .
A hypothesis test is conducted at a particular significance level to test whether � is greater than 250 .
( a ) ( i ) Write down the null hypothesis of the test .
( ii ) Write down the alternative hypothesis of the test .
The null hypothesis is rejected if it is observed that the mean of a sample of size 25 is greater than 251 .
( b ) Find the probability that a Type I error is made .
The actually value of � is 250.5 .
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( c ) Find the probability that a Type II error is made .
Another random variable Y is normally distributed with parameter � , where � represents the mean value of a new population . The population variance is 11 .
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A hypothesis test is conducted at a 5 % significance level to test whether � is not equal to 300 . It is observed that the mean is 279 in a random sample of size 15 .
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( d ) Write down the critical region for testing � , giving the answer in the form ( Y �a or Y � b) .
( e ) State the conclusion of the test with a reason .
Assume that the level of significance is changed to 10 %.
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( f ) Write down the new critical region for testing � , giving the answer in the form ( Y �a or Y � b) .
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