The weights of coconuts from a plantation have mean 3 and standard deviation 1 (

Question

The weights of coconuts from a plantation have mean 3 and standard deviation 1 (pounds). 1. One coconut is selected randomly. Find the probability that it weighs more than 3.5 pounds. To do this, first assume that the population of weights is normally distributed. 2. Fifty coconuts are selected randomly. Find the probability that their average weight is more than 3.5 pounds. (Do not assume any longer that the population of weights is normally distributed.) 3. Find the probability the same fifty coconuts together weigh more than 175 pounds.

Explanation / Answer

1)

P(weight more than 3.5 pounds)=P(X>3.5)=P(Z>(3.5-3)/1)=P(Z>0.5)=0.3085

2)

here standrd error of mean =standard deviaiton/sqrt(n)=1/sqrt(50)

hence from central limit theorum and normal approximation

P(Xbar>3.5)=P(Z>(3.5-3)/(1/sqrt(50))=P(Z>3.54)=0.0002

3)

here expeced mean weight of 50 coconuts=50*3=150

and std deviaiton=1*sqrt(50)=7.071

hence P(X>175)=P(Z>(175-150)/7.071)=P(Z>3.54)=0.0002

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