Exercise 68
1 . The random variable X is defined such that E ( X ) � 300 and Var ( X ) � 4.5 . A random
sample of 180 observations is selected from the distribution of X . Let X be the mean of the sample . By using the central limit theorem ,
( a ) write down E ( X ) ;
( b ) find Var ( X ) ;
( c ) find P ( X � 299.85 ) .
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2 . The random variable X is defined such that E ( X ) �� 2 and Var ( X ) � 8. A random
sample of 32 observations is selected from the distribution of X . Let X be the mean of the sample . By using the central limit theorem ,
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( a ) write down E ( X ) ;
( b ) find the standard deviation of X ;
( c ) find P ( X � 1.5 ) .
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3 . The random variable X is defined such that E ( X ) � 5 and Var ( X ) � 1.6 . A random
sample of 50 observations is selected from the distribution of X . Let X be the mean of the sample . By using the central limit theorem ,
( a ) write down E ( X ) ;
( b ) find Var ( X ) .
Another random variable Y is defined such that E ( Y) �� 5 and Var ( Y) � 0.8 . A new random sample of 50 observations is selected from the distribution of Y . Let Y be the mean of the new sample . It is given that X and Y are independent .
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( c ) Find P ( X �Y
� 0.1 ) . [ 3 ]
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