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The following is a problem for A-level test that I don't know how to solve.

The weight of a pig is normally distributed with mean 375 g and standard deviation 22 g. The weight of a rabbit is normally distributed with mean 425 g and standard deviation 25 g.

i) Five pigs and four rabbits are randomly chosen. Find the probability that the pigs weigh more than the rabbits.

ii) Find the probability that the average weight of a sample of 10 rabbits will exceed the average weight of a sample of 8 pigs by at least 39 g.

Can someone help me? Thanks a lot!

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Guide:

i)

Let $P_i$ be normally distributed for $i=1,2,3,4,5$ with the parameters that you mention for a pig.

Let $R_i$ be normally distributed for $i=1,2,3,4$ with the parameters that you mention for a rabbit.

Then $X:=P_1+P_2+P_3+P_4+P_5-R_1-R_2-R_3-R_4$ has normal distribution.

Now find the parameters of this distribution and then find $P(X>0)$.

ii)

If $\overline P$ denotes the average weight of $8$ pigs, and $\overline R$ denotes the average weight of $10$ rabbits.

Then $Y:=\overline R-\overline P$ has normal distribution.

Now find the parameters of this distribution and then find $P(Y>39)$ or equivalently $P(Y-39>0)$.

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    $\begingroup$ @p32fr4 Thank you, I overlooked the $39$. $\endgroup$ – drhab Dec 18 '18 at 10:07
  • $\begingroup$ Understood! Thanks a lot! $\endgroup$ – mapping Dec 18 '18 at 11:22
  • $\begingroup$ You are welcome. $\endgroup$ – drhab Dec 18 '18 at 11:23

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