Question

Slices of pizza for a certain brand of pizza have a mass that is approximately normally distributed with a mean of 66.8 grams and a standard deviation of 1.82 grams. Round answers to three decimal places.

a) For samples of size 22 pizza slices, what is the standard deviation for the sampling distribution of the sample mean?

b) What is the probability of finding a random slice of pizza with a mass of less than 66.3 grams?

c) What is the probability of finding a 22 random slices of pizza with a mean mass of less than 66.3 grams?

d) What sample mean (for a sample of size 22) would represent the bottom 15% (the 15th percentile)? grams

Answer 1

the PDF of normal distribution is = 1/σ * √2π * e ^ -(x-u)^2/ 2σ^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
equation of the normal curve is ( Z )= x - u / sd ~ N(0,1)
mean ( u ) = 66.8
standard Deviation ( sd )= 1.82
a.
For samples of size 22 pizza slices, the standard deviation for the sampling distribution of the sample mean
standard deviation of the sampling distribution = 1.82/sqrt(22) = 0.388
b.
the probability of finding a random slice of pizza with a mass of less than 66.3 grams
P(X < 66.3) = (66.3-66.8)/1.82
= -0.5/1.82= -0.275
= P ( Z <-0.275) From Standard Normal Table
= 0.392
c.
the probability of finding a 22 random slices of pizza with a mean mass of less than 66.3 grams
the PDF of normal distribution is = 1/σ * √2π * e ^ -(x-u)^2/ 2σ^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
equation of the normal curve is ( Z )= x - u / sd/sqrt(n) ~ N(0,1)
mean of the sampling distribution ( x ) = 66.8
standard Deviation ( sd )= 1.82/ Sqrt ( 22 ) =0.388
sample size (n) = 22
P(X < 66.3) = (66.3-66.8)/1.82/ Sqrt ( 22 )
= -0.5/0.388= -1.2886
= P ( Z <-1.2886) From Standard NOrmal Table
= 0.099
d.
sample mean (for a sample of size 22) would represent the bottom 15% (the 15th percentile) grams
P ( Z < x ) = 0.15
Value of z to the cumulative probability of 0.15 from normal table is -1.036
P( x-u/s.d < x - 66.8/1.82 ) = 0.15
That is, ( x - 66.8/1.82 ) = -1.036
--> x = -1.036 * 1.82 + 66.8 = 64.914