Question

You are planning to take to a trip to Montreal, Canada during the month of April and you want to bring clothing that is appropriate for the weather. The daily high temperature X in degrees Celsius in Montreal during April has expected value E(X) = 10.3oC with a standard deviation SD(X) = 3.5oC. You want to convert these Celsius temperatures to oF (degrees Fahrenheit). The conversion of X into degrees Fahrenheit Y is Y = (9/5)X + 32.

1. What is E(Y), the expected daily high in Montreal during April in degrees Fahrenheit?
2. What is SD(Y), the standard deviation of the daily high temperature in Montreal during April in degrees Fahrenheit?

Answer 1

1. E(Y) = 50.54°F

2. SD(Y) = 11.34°F

Explanation:

We are given that The daily high temperature X in degrees Celsius in Montreal during April has expected value E(X) = 10.3°C with a standard deviation SD(X) = 3.5°C.

The conversion of X into degrees Fahrenheit Y is Y = (9/5)X + 32.

(1) Y = (9/5)X + 32

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

= (9/5) * E(X) + 32 {
`\because` expectation of constant is constant}

= (9/5) * 10.3 + 32 = 50.54

Therefore, E(Y), the expected daily high in Montreal during April in degrees Fahrenheit is 50.54°F .

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

SD(Y) = SD((9/5)X + 32) = SD((9/5)X) + SD(32)

=
`(9/5)^(2)` * SD(X) + 0 {
`\because` standard deviation of constant is zero}

=
`\beacuse`
`(9/5)^(2)` * 3.5 = 11.34°F

Therefore, SD(Y), the standard deviation of the daily high temperature in Montreal during April in degrees Fahrenheit is 11.34°F .