1. A biologist has a computer file showing 1,000 weights of rabbits in a certain
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Question
1. A biologist has a computer file showing 1,000 weights of rabbits in a certain study. These range from 105 (grams) to 4,500 (grams). By accident, the largest weight in the file gets changed to 45,000.
Does this affect the mean? If so, by how much? Find the exact error.
Does this affect the median? If so, by how much? Find the exact error.
2. A sample of 100 length measurements has a mean of 106 mm and a standard deviation of 14 mm. Some of the observations are 98, 87, 115, ...
a) Find the total sum of squared deviations
b)Suppose the same data set ( x=106, sd=14, n=10) has each value duplicated in the sample, thus creating a new sample of 200 measurements with some observations:
b) 98, 98, 87, 87, 115, 115, ... Find the mean of the new sample. How much does it change from the original mean?
c) Find the standard deviation of the new sample. How much does it change from the original standard deviation?
3. Suppose you have two sample means x and y : x from data x1,x2,....xn and y from data y1 ,y2,.... ym (n and m may be different). What would be the new mean when two sets of data are combined? Denote the new mean of the combined data using onlyn,m, x, and y.
4. Suppose you have a sample mean and a standard deviation s from data . Suppose you add 10 to each value in the data set.
a) Does this affect the mean? If so, by how much?
b) Does this affect the standard deviation? If so, by how much?
c)Does this affect the variance? If so, by how much?
5. Suppose you have sample mean x and standard deviation s from data . Suppose you multiply 10 to each value.
a) Does this affect the mean? If so, by how much?
b) Does this affect the standard deviation? If so, by how much?
c) Does this affect the variance? If so, by how much?
Explanation / Answer
A.1) number of values=1000
initial mean=m
initial sum of observations=1000*m
new sum of observations=1000*m-4500+45000
new mean=(1000*m+40500)/1000
new mean=m+40.5
the new mean is increased by 40.5
the exact error is 40.5
the median is average of 500th and 501 value
as the largest value is changed by accident and not the middle ones,
there won’t be any affect on the median
the median remains the same
A.2) SD=Sqrt(SSD/n)
14=Sqrt(SSD/100)
196=SSD/100
SSD=19600
total sum of squared deviations is 19600
given, each value is duplicated
old mean=(x1+x2+……….+xn)/100
new mean=(x1+x1+x2+x2+……..+xn+xn)/200
new mean=2*(x1+x2+…….+xn)/200
new mean=old mean
hence there is no affect on mean
similarly there wont be any affect on standard deviation
A.3) we have,
Xn=mean*no of observations
Xn=x*n
Ym=y*m
now we need to find combined mean
combined mean={(x1+x2+……….+xn)+(y1+y2+……..+ym)}/(n+m)
combined mean=(x*n+y*m)/(n+m)
A.4) sample mean=M
standard deviation=SD
suppose we add 10 to each value
new mean={(X1+10)+(X2+10)+……….(Xn+10)}/n
new mean=(X1+X2+……..Xn/n)+(10*n/n)
new mean=M+10
hence the mean is increased by 10
new SD=Sq rt {(X1+10-M-10)^2+(X2+10-M-10)^2+…….. (Xn+10-M-10)^2}/n
new SD= Sq rt{(X1-M)^2+(X2-M)^2+……..+(Xn-M)^2}/n
which is similar to old SD
hence there is no affect on Standard deviation
same applies to variance as well
there is no affect on Standard deviation
A.5) sample mean=M
standard deviation=SD
suppose we multiply 10 to each value
new mean=(10*X1+10*X2+…………10*Xn)/n
new mean=10*(X1+X2+………….Xn)/n
new mean=10*old mean
hence mean is increased by 10 times
new SD= Sq rt{(10X1-10M)^2+(10X2-10M)^2+………..(10Xn-10M)^2}/n
new SD=10*old SD
hence, the new Standard deviation is increased by 10 times
similarly, the new variance is increased by 100 times
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