Exploring The Central Limit Theorem and Sample Proportions in Fathom
Find the true proportion of 4 year old pennies from the population of pennies
p = .053
This value is called a parameter
since is describes the population
Use samples of size n = 10 to plot part of the sampling distribution for
samples of size n = 10. In each sample, calculate the proportion of 4 year old
pennies. Use 1000 samples of size n = 10.
1.) Paste your graph below. Also paste the summary table below.
2.) Theoretical mean and standard deviation for sampling distributions of
sample proportions:
p = the proportion of 4 year old pennies in a random sample of 10 pennies
µ p = p = .053
σ p =
p(1 − p)
.053(1 − .053)
=
= .071
n
10
3.) What is the probability that a random sample of 10 pennies will contain
less than 3% of 4 year old pennies in the sample? P( p < .03 )
SINCE THE SAMPLING DISTRIBUTION FOR p IS SKEWED RIGHT,
WE CANNOT USE NORMAL CURVE CALCULATIONS TO FIND THIS
PROBABILITY
Conditions
For
normality
1.)
np ≥ 10
(10)(.053) = .53 is not greater than or equal to 10,
so the first condition for normality is not met
2.) n(1-p) ≥ 10
10(1-.053) = 9.47 is not greater than or equal to 10, so
the second condition for normality is not met
Exploring The Central Limit Theorem and Sample Proportions in Fathom
Use samples of size n = 100 to plot part of the sampling distribution for
samples of size n = 100. In each sample, calculate the proportion of 4 year
old pennies. Use 1000 samples of size n = 100.
1.) Paste your graph below. Also paste the summary table below.
2.) Theoretical mean and standard deviation for sampling distributions of
sample proportions:
p = the proportion of 4 year old pennies in a random sample of 100 pennies
µ p = p = .053
p(1 − p)
.053(1 − .053)
=
= .022
n
100
3.) What is the probability that a random sample of 100 pennies will contain
less than 3% of 4 year old pennies in the sample? P( p < .03 )
SINCE THE SAMPLING DISTRIBUTION FOR p IS SKEWED RIGHT,
σ p =
WE CANNOT USE NORMAL CURVE CALCULATIONS TO FIND THIS
PROBABILITY
Conditions
For
normality
1.)
np ≥ 10
(100)(.053) = 5.3 is not greater than or equal to 10,
so the first condition for normality is not met
2.) n(1-p) ≥ 10
100(1-.053) = 9.46 is greater than or equal to 10,
so the second condition for normality is not met
SINCE BOTH CONDITIONS ARE NOT MET, WE CANNOT ASSUME
NORMALITY FOR THE p DISTRIBUTION
Exploring The Central Limit Theorem and Sample Proportions in Fathom
Use samples of size n = 200 to plot part of the sampling distribution for
samples of size n = 200. In each sample, calculate the proportion of 4 year
old pennies. Use 1000 samples of size n = 200.
1.) Paste your graph below. Also paste the summary table below.
2.) Theoretical mean and standard deviation for sampling distributions of
sample proportions:
p = the proportion of 4 year old pennies in a random sample of 200 pennies
µ p = p = .053
p(1 − p)
.053(1 − .053)
=
= .016
n
200
3.) What is the probability that a random sample of 200 pennies will contain
less than 3% of 4 year old pennies in the sample? P( p < .03 )
Conditions for normality :
np ≥ 10
200(.053)=10.6 is greater than 10
BOTH MET
n(1-p) ≥ 10
200(1.-.053)=189.4 is greater than 10
σ p =
⎛
(.053)(1 − .053) ⎞
BOTH assumptions met therefore p  N ⎜ .053,
⎟
200
⎝
⎠
BY THE CENTRAL LIMIT THEOREM
WE CAN CALCULATE P( p < .03 ) = .0733
There is a 7.33% chance of obtaining a sample of 200 pennies in which
less than 3% of the pennies are 4 years old.