Question

In: Statistics and Probability

40 independent measurements of the boiling point of a certain liquid were found to have a...

40 independent measurements of the boiling point of a certain liquid were found to have a sample average of 950C, and sample variance of 20C. The formula for converting from a Celsius temperature x to a Fahrenheit temperature y is y=9x5+32. Thus, the sample mean and sample variance of the measurements (on the Fahrenheit scale) are, respectively:

203 0F and 38.48 0F

203 0F and 6.48 0F

171 0F and 38.48 0F

171 0F and 2.54 0F

203 0F and 2.54 0F

Solutions

Expert Solution

Given, n = 40, Mean = E(x) = 95, Sample variance = Var(x) = 2

y = (9/5)*x+32

E(y) = E[(9/5)*x+32]

Using summation rule of mean E(X+c) = E(X)+c, we get

E(y) = E[(9/5)*x] + 32

E(y) = E[(9/5)*x] + 32

Using multiplication rule of mean E(cX ) = cE(X), we get

E(y) = (9/5)*E(x) + 32

Substituting the value of E(x), we get

E(y) = (9/5)*95 + 32

E(y) = 203

Var(y) = Var[(9/5)*x+32]

Using summation rule of variance Var(X+c) = Var(X), we get

Var(y) = Var[(9/5)*x]

Using multiplication rule of mean Var(cX ) = c2Var(X), we get

Var(y) = (9/5)2Var(x)

Substituting the value of Var(x), we get

Var(y) = 81*2/25

Var(y) = 6.48

Thus, the sample mean and sample variance of the measurements (on Fahrenheit scale) are, respectively 203 0F and 6.48 0F


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