Question

Jars of relish have a normal distribution with a mean weight of 8.12 ounces and a standard deviation of 0.09 ounce. A.) If one jar of relish is selected at random, what is the probability that the weight of that jar is more than 8.20 ounces? B.) If 10 jars of relish are selected at random, what is the probability that the mean of these 10 jars is more than 8.20 ounces.

Answer 1

a)

X ~ N ( µ = 8.12 , σ = 0.09 )
We covert this to standard normal as
P ( X < x) = P ( (Z < X - µ ) / σ )
P ( X > 8.2 ) = P(Z > (8.2 - 8.12 ) / 0.09 )
= P ( Z > 0.89 )
= 1 - P ( Z < 0.89 )
= 1 - 0.8133
= 0.1867

b)

X ~ N ( µ = 8.12 , σ = 0.09 )
P ( X > 8.2 ) = 1 - P ( X < 8.2 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 8.2 - 8.12 ) / ( 0.09 / √ ( 10 ) )
Z = 2.81
P ( ( X - µ ) / ( σ / √ (n)) > ( 8.2 - 8.12 ) / ( 0.09 / √(10) )
P ( Z > 2.81 )
P ( X̅ > 8.2 ) = 1 - P ( Z < 2.81 )
P ( X̅ > 8.2 ) = 1 - 0.9975
P ( X̅ > 8.2 ) = 0.0025