Please show all steps 6. Complete the following steps: A. Using a random number

Question

Please show all steps

6. Complete the following steps: A. Using a random number generator, create Set 1 of normally distributed data with 1000 values with mean 50 and standard deviation 15. If doing this with an Excel command, set the seed to 1 Again, using a random number generator, create Set 2 of normally distributed data with 1000 values with mean 100 and standard deviation 20. If doing this with an Excel command, set the seed to 2. Now create Set 3 of values by adding Set 1 and Set 2 in a row-by-row fashion. This set is also normally distributed. Calculate the actual mean and standard deviation of each of the three sets of data. B. Calculate the sum of the means of Set 1 and Set 2 and compare it to the calculated mean of Set 3. Calculate the square root of the sum of the squares of the standard deviations of Set 1 and Set 2 and compare it to the calculated standard deviation of Set 3. Comment on both of these comparisons C. Using Set 1 and its calculated actual mean and standard deviation, determine the percentages of values that are within plus and minus one, plus and minus two, and plus and minus three standard deviations of the mean. The Histogram tool in Excel will be useful for this. Tabulate your values and compare your values with the expected values predicted by statistics.

Explanation / Answer

A.

Set 1: mean=50.14481, SD=15.13969

Set 2: mean=100.3421, SD=20.6543

Set 3: mean=150.4869, SD=24.95788

B.

Sum of Means of Set 1 and Set 2=50.14481+100.3421=150.4869 which same as the mean of Set 3.

The square root of the sum of the squares of the SDs of Set 1 and Set 2=sqrt(15.13969^2+20.6543^2)

=25.60879 which greater than the standard deviation of Set 3.

Hence Var(Set 1)+Var(Set 2)>Var(Set 1+Set 2)=Var(Set 1)+Var(Set 2)+2cov(Set 1, Set 2)

Therefore this is happen because cov(Set 1, Set 2)<0

C. Percentage values that are within plus and minus expected value=68.3%

Percentage values that are within plus and minus 2=95.8%, expected value=95.4%

Percentage values that are within plus and minus 3=99.9%, expected value=99.7%

R code:

#Set 1:
x=rnorm(1000,50,15)# generate 1000 random no.s from N(50,15)
#Set 2:
y=rnorm(1000,100,20)# generate 1000 random no.s from N(100,20)
# Set 3:
z=x+y
mean(x)
sd(x)
mean(y)
sd(y)
mean(z)
sd(z)
u=(x-mean(x))/sd(x)# convert x to z-score
p1=(length(subset(u,u>=-1&u<=1))/1000)*100 #Percentage values that are within plus and minus one
p2=(length(subset(u,u>=-2&u<=2))/1000)*100#Percentage values that are within plus and minus two
p3=(length(subset(u,u>=-3&u<=3))/1000)*100#Percentage values that are within plus and minus three
e1=(pnorm(1)-pnorm(-1))*100#percentage expected values within plus and minus one
e2=(pnorm(2)-pnorm(-2))*100#percentage expected values within plus and minus two
e3=(pnorm(3)-pnorm(-3))*100#percentage expected values within plus and minus three

p1

p2

p3

e1

e2

e3

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