A sample of 30 distance scores measured in yards has a mean of 10, a variance of

Question

A sample of 30 distance scores measured in yards has a mean of 10, a variance of 9, and a standard deviation of 3. (a) you want to convert all your distances from yards to feet, so you multiply each score in the sample by 3. what are the new mean, variance, and standard deviation?(b) you then decide that you only want to look at the distance past a certain point. thus, after multiplying the original scores by 3, you decide to subtract 4 feet from each of the scores. now what are the new mean, variance and standard deviation?

Explanation / Answer

Let the measured value in yards be X. Then we are given that:

E(X) = 10 because the mean is 3. and STD(X) = 3

a) Here we are given that if the measured value is in feet, then Y = 3X

Therefore, new mean: E(Y) = E(3X) = 3E(X) = 3*10 = 30

Therefore 30 is the new mean.

Similarly, new standard deviation: SD(Y) = SD(3X) = 3SD(X) = 3*3 = 9

Therefore 9 is the new standard deviation.

b) Now here the new variable is given as: Z = 3X - 4 = Y - 4

Therefore new mean is computed as: E(Z) = E(Y - 4) = E(Y) - 4 = 30 - 4 = 26

Therefore the new mean is 26

Now computing the new standard deviation:

SD(Z) = SD(Y - 4)

Note that standard deviation of a constant is 0. Therefore we get:

SD(Z) = SD(Y - 4) = SD(Y) = 9

Therefore the standard deviation is still 9

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