Transcript 7.2 Notes - morgansmathmarvels

7.2 Notes
Central Limit Theorem
Theorem 7.1 – Let x be a normal distribution with mean μ and standard
deviation σ.
Let x be a sample mean corresponding to random sample of size n.
http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/
Ex. 1 Pinedale fishing pond has trout whose lengths are normally distributed
with mean μ = 10.2 in. and standard deviation σ = 1.4 in.
a) What is the probability that a single trout chosen at random is between 8 and
12 in.?
b) What is the probability that x of 5 trout chosen at random is between 8 and
12 in.?
c) Are the two values the same? Why or why not?
Central Limit Theorem – If x possesses any distribution with mean μ and
standard deviation σ, then the sample mean x based on random sample size of
n will have a distribution that approaches normal with  x   and   
x
n
as n approaches
.
∞
The essential question is “How large does n have to be to approximate
normal?”
http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/
Ex. 2 Let x be the bacteria count per ml of raw milk with μ = 2500 and σ = 300.
A health inspector takes 42 random samples of the raw milk produced each
day. At the end of each day, the bacteria count in 42 samples is averaged to
obtain x .
a) What can we say about x distribution?
b) What is the probability that the average bacteria count for one day x is
between 2350 and 2650 bacteria per ml?
c) What is the probability that the average bacteria count for one day x is
greater than 2650?
*Ex. 3 Taxi and takeoff time for commercial jets has μ = 8.5 min. and σ = 2.5
min. Assume the jets are lined up on a runway so that one taxis and takes off
immediately after the other. What is the probability that for 36 jets on a given
runway, the total taxi and takeoff time will be less than 320 min.?
Assignment
Day 1 p. 306 #4, 9, 11 , 15, 19
Day 2 p. 306 # 6, 8, 12, 13, 16, 20