Question

1. Wayne’s favourite donut at Tim Hortons is the sour cream glazed donut. Tim Hortons claims that each of these donuts contains 320 calories. Suppose that the actual number of calories in a sour cream glazed donut varies according to a normal distribution with a mean of 317 calories and a standard deviation of 22 calories. Let X represent the number of calories in a randomly selected sour cream glazed donut from Tim Hortons.

(a) What is the probability that the next sour cream glazed donut Wayne consumes will contain at least 300 calories?

(b) What proportion of all sour cream glazed donuts contain between 310 and 330 calories?

(c) Tim Hortons can guarantee that only 1.5% of all sour cream glazed donuts contain more than how many calories?

Answer 1

Part a)

X ~ N ( µ = 317 , σ = 22 )
P ( X >= 300 ) = 1 - P ( X < 300 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 300 - 317 ) / 22
Z = -0.77
P ( ( X - µ ) / σ ) > ( 300 - 317 ) / 22 )
P ( Z > -0.77 )
P ( X >= 300 ) = 1 - P ( Z < -0.77 )
P ( X >= 300 ) = 1 - 0.2206
P ( X >= 300 ) = 0.7794

Part b)

X ~ N ( µ = 317 , σ = 22 )
P ( 310 < X < 330 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 310 - 317 ) / 22
Z = -0.32
Z = ( 330 - 317 ) / 22
Z = 0.59
P ( -0.32 < Z < 0.59 )
P ( 310 < X < 330 ) = P ( Z < 0.59 ) - P ( Z < -0.32 )
P ( 310 < X < 330 ) = 0.7224 - 0.3745
P ( 310 < X < 330 ) = 0.3479

Part c)

X ~ N ( µ = 317 , σ = 22 )
P ( X > ? ) = 1 - P ( X < ? ) = 1 - 0.015 = 0.985
Looking for the probability 0.985 in standard normal table to calculate critical value Z = 2.17
Z = ( X - µ ) / σ
2.17 = ( X - 317 ) / 22
X = 364.74
P ( X > 364.74 ) = 0.015