2 . This question aims at investigating the weights of a population of crabs .
Alex owns a crab farm in Vietnam . During an interview he claims that the weights W of the crabs in his crab farm is normally distributed with mean 300 grams and standard deviation 10 grams .
( a ) |
( i ) |
Write down the probability that a randomly selected crab from |
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the crab farm weighs less than 292 grams . |
A sample of a dozen of crabs is randomly selected .
( ii ) Find the probability that the sample mean weight of this sample is less than 292 grams .
[ 4 ]
A crab cage is designed such that the weight limit of the cage is 6.1 kilograms . Alex is trying to carry twenty crabs by the cage .
( b ) ( i ) Write down the expected total weight of the twenty crabs .
( ii ) Find the variance of the total weight of the twenty crabs .
( iii ) Hence , find the probability that the total weight exceeds the weight limit of the cage .
[ 5 ] Alex would like to conduct a hypothesis test at a 5 % significance level to test whether there is a negative correlation between the weight of a crab with its maximum walking speed . The table below shows the data of a sample of five crabs :
Weight ( W g ) 280 286 292 298 304 Maximum walking speed ( V cm / s ) 86 66 66 48 49
It is also assumed that the maximum walking speeds of crabs are normally distributed . Let � be the product moment correlation coefficient of the two variables .
( c ) ( i ) Write down the null hypothesis of the test .
( ii ) Write down the alternative hypothesis of the test .
( iii ) Find the p -value .
( iv ) Hence , state the conclusion of the test with a reason . [ 6 ]
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