You are studying the temperature of Victoria in the winter months and you find i

Question

You are studying the temperature of Victoria in the winter months and you find it has an average temperature of 5 C with a variance of 5 C. You wish to present your results at a conference down in the United States. You know the relationship between Celsius and Fahrenheit is given by the equation Y = 9 5 X + 32 where X= temperature in C and Y= temperature in F .

What is the expected temperature value of Victoria in degrees Fahrenheit and what is the standard deviation of Victoria’s temperature in degrees Fahrenheit?

Explanation / Answer

Here ,

The equation is given as

Y = 95X+32

But we know that the relationship between Celcius and Fahrenheit is

Y = (9/5) X+32

We SOLVE with both the equations for clarity,

Given,

E(X)=5°C ( average temperature of 5 C) Var(X)=5°C (variance of 5 C)

We know that 1. E(aX+b)=a*E(X)+b where X is a random variable and a,b are real constants.

2. Var(aX+b)=a2*Var(X) where a is real constant

Case 1: Y = 95X+32

Now, E(Y)=95*E(X)+32

E(Y)=95*E(X)+32

E(Y)=95*5+32

=507°F

Var(Y)=952*Var(X)

=9025*5

=45125

standard deviation of Y= [Var(X)]0.5

=(45125)0.5

   =212.4264579°F

Case 2: Y = (9/5)X+32

E(Y)=(9/5)*E(X)+32

E(Y)=(9/5)*5+32

=41°F

Var(Y)=(9/5)2*Var(X)

=(1.8)2*5

=16.2

Standard deviation of Y= [Var(X)]0.5

=(16.2)0.5

=4.024922359°F

So the answer is

1. When Y= 95 X + 32

Then expected temperature value of Victoria in Fahrenheit is 507°F and the standard deviation of Victoria's temperature would be 212.4264579°F.

1. When Y= 95 X + 32

Then expected temperature value of Victoria in Fahrenheit is 507°F and the standard deviation of Victoria's temperature would be 212.4264579°F.

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