It Makes Cents
Use Fathom to graph a histogram of the distribution of the ages of the
pennies. Paste your graph below.
x = the age of one randomly selected penny
u x = 19.22
σ x = 12.62
What is the shape of the distribution above? SKEWED TO THE RIGHT
Can we use normal curve calculations to compute the probability that the
age of a randomly selected penny will exceed 10 years? P(x>10)
NO. SINCE THE X-DISTRIBUTION IS SKEWED TO THE RIGHT, WE
CANNOT USE NORMAL CURVE CALCULATIONS TO FIND THIS
PROBABILITY
Use Fathom to graph a histogram of the sampling distribution of sample
means for size n=2. Paste your graph below.
Note: The above graph is only part of the sampling distribution for
samples of size n = 2. There are only 1000 sample means graphed
here. The true sampling distribution for samples of size n = 2 is
comprised of (2279 choose 2)= 2,595,781 samples.
Theoretical mean and standard deviation of the sampling distribution:
x = the mean age of two randomly selected pennies
u x = µ x = 19.22
σx =
σ x 12.62
=
= 8.92
n
2
What is the shape of the distribution above? SKEWED RIGHT.
Can we use normal curve calculations to compute the probability that the
average age of two randomly selected pennies will exceed 13 years?
P( x >13)
NO. SINCE THE x DISTRIBUTION IS SKEWED TO THE RIGHT,
WE CANNOT USE NORMAL CURVE CALCULATIONS TO FIND THE
ABOVE PROBABILITY.
Use Fathom to graph a histogram of the sampling distribution of sample
means for size n=4. Paste your graph below.
Theoretical mean and standard deviation of the sampling distribution:
x = the mean age of four randomly selected pennies
u x = 19.22
σx =
σ x 12.62
=
= 6.31
n
4
What is the shape of the distribution above? SKEWED RIGHT
Can we use normal curve calculations to compute the probability that the
average age of four randomly selected pennies will exceed 13 years?
P( x >13)
NO. SINCE THE x DISTRIBUTION IS SKEWED TO THE RIGHT,
WE CANNOT USE NORMAL CURVE CALCULATIONS TO FIND THE
ABOVE PROBABILITY.
Use Fathom to graph a histogram of the sampling distribution of sample
means for size n=30. Paste your graph below.
Theoretical mean and standard deviation of the sampling distribution:
x = the mean age of 30 randomly selected pennies
u x = µ x = 19.22
σx =
σ x 12.62
=
= 2.30
n
30
What is the shape of the distribution above? APPROXIMATELY
NORMAL
Can we use normal curve calculations to compute the probability that the
average age of thirty randomly selected pennies will exceed 13 years?
12.62 ⎞
⎛
YES. x  N ⎜ 19.22,
⎟ BECAUSE n ≥ 30 by the CENTRAL LIMIT
⎝
30 ⎠
THEOREM
P( x >13)=.9977
1.)
What is the probability that one penny chosen at random from
the population will be at least 20 years old?
x = the age of one randomly selected penny
P(x ≥ 20)=
We cannot calculate this probability because the
x-distribution is not normal
2.)
What is the probability that an SRS of n=50 pennies will have a
mean age of 20 or more years old?
x = the mean age of 50 randomly selected pennies
P( x ≥ 20) = .3229
We can compute this probability
12.62 ⎞
⎛
since x  N ⎜ 19.22,
⎟ because n is large by the Central
⎝
50 ⎠
Limit Theorem