Hi I got A! But B and C I need help with! In an agricultural experiment, a large
ID: 3208196 • Letter: H
Question
Hi I got A! But B and C I need help with!
In an agricultural experiment, a large field of wheat was divided into many plots (each plot being 7 x 100 ft) and the yield of grain was measured for each plot. These plot yields approximately followed a normal distribution with mean 70 lbs. and SD 8 lbs.
a.
Let y5 represent the mean yield rate of 5 plots chosen at random from the field. According to CLT, what is the mean and standard deviation of y5? And find the probability that y5 is greater than 80 lbs.
Mean ( u ) =70
Standard Deviation ( sd )= 8/ Sqrt(n) = 3.5777
Number ( n ) = 5
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)
P(X > 80) = (80-70)/8/ Sqrt ( 5 )
= 10/3.578= 2.7951
= P ( Z >2.7951)
= 0.0026
b.
Let y5 represent the mean yield rate of 100 plots chosen at random from the field, According to CLT, what is the mean and standard deviation of y5? And find the probability that y5 is greater than 80 lbs.
c.
Does the sample size affect the use of the Central Limit Theorem in b) and c)? Why or why not?
Explanation / Answer
b)
Mean ( u ) =70
Standard Deviation ( sd )= 8/ Sqrt(n) = 8/10 = 0.8
Number ( n ) =100
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)
P(X > 80) = P ( Z >12.5) = 0
C) The sample size does affect the use of the Central Limit Theorem because the CLT should only be used with sample sizes that are large. since for small sample sizes , the shape of the distributions are far from normal. In order to use CLT, the sample size should atleast be 25. The central limit theorem says that the sampling distribution approximates a bell shape given that the sample is large enough.