In: Statistics and Probability
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
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