Statistic 8.1
Inference for a Single Proportion
X
The sample proportion of successes ^p= estimates the
n
unknown population proportion p
o “success” to represent the characteristics of interest
if the population is much larger than the sample , the count X
has approximately the binomial distribution B(n, p)
if the sample size n is very small, we must base tests and
confidence intervals for p on the binomial distributions
Large-sample confidence interval for a single proportion
μ ^p= p and σ ^p =
√ p ( 1− p )
n
approximately 95% of the time
of the unknown population proportion p
^p will be within 2
√ p ( 1− p )
n
the standard deviation σ ^p depends upon the unknown
parameter p
X
The sample proportion is ^p= where X is the number of
n
successes
The standard error of ^p is SE ^p=
n √
^p ( 1− ^p )
and
The margin of error for confidence level C is m=z * SE ^p
where the critical value z* is the value for the standard Normal
density curve with area C between –z* and z*
An approximate level C confidence interval for p is ^p ± z *
√ ^p (1− ^p )
n
Use this interval for 90*, 95%, or 99% confidence when the
number of successes and the number of failures are both at
least 15
Remember, that the margin of error in any confidence interval
include only random sampling error
The plus four confidence interval for a single proportion
When the number of successes and the number of failures are
not at least 15
The estimator of the population proportion based on this plus
X +2
four rule is ^p=
n+ 4
, Plus four estimate ~
p ; mean p and standard deviation
√ p (1− p)/(n+ 4)
Significance test for a single proportion
Significance test = we assume that the null hypothesis is true
To test the hypothesis Ho: p = p0 compute the z statistic
^p− p0
z=
o
√ p0 (1− p 0)
n
H a : p> p 0 is P ( Z ≥ z )
o H a : p< p 0 is P ( Z ≤ z )
o H a : p ≠ p0 is 2 P ( Z ≥|z|)
we do not often use significance tests for a single proportion,
because it is uncommon to have a situation where there is a
precise p0 that we want to test
Choosing a sample size
because we don’t know the value of ^p until we gather the
data, we must guess a value to use in the calculations – the
guessed value is p*
there are two common ways to get p*
o 1. Use the sample estimate from a pilot study or from
similar studies done before
o use p*=0.5. because the margin of error is largest when
^p =0.5; it is a safe choice no matter what the data
later show
z¿ 2
n=( ) p*(1 – p*)
m
o here z* is the critical value for confidence C, and p* is a
guessed value for the proportion of successes in the
future sample
¿
the sample size required when p* = 0.5 is n= 1 ¿ z*/m)2
4
the value of n obtained by this method is not particularly
sensitive to the choice of p* when p* is fairly close to 0.5
e.g. will the proposed sample size of n = 200 be adequate to
provide Rec Sports with the needed information? To address
this we calculate a 95% confidence interval for various values
of ^p
the margins of error will always be the same for ^p and 1 -
^p (p.479)
p. 478 Example 8.7
Inference for a Single Proportion
X
The sample proportion of successes ^p= estimates the
n
unknown population proportion p
o “success” to represent the characteristics of interest
if the population is much larger than the sample , the count X
has approximately the binomial distribution B(n, p)
if the sample size n is very small, we must base tests and
confidence intervals for p on the binomial distributions
Large-sample confidence interval for a single proportion
μ ^p= p and σ ^p =
√ p ( 1− p )
n
approximately 95% of the time
of the unknown population proportion p
^p will be within 2
√ p ( 1− p )
n
the standard deviation σ ^p depends upon the unknown
parameter p
X
The sample proportion is ^p= where X is the number of
n
successes
The standard error of ^p is SE ^p=
n √
^p ( 1− ^p )
and
The margin of error for confidence level C is m=z * SE ^p
where the critical value z* is the value for the standard Normal
density curve with area C between –z* and z*
An approximate level C confidence interval for p is ^p ± z *
√ ^p (1− ^p )
n
Use this interval for 90*, 95%, or 99% confidence when the
number of successes and the number of failures are both at
least 15
Remember, that the margin of error in any confidence interval
include only random sampling error
The plus four confidence interval for a single proportion
When the number of successes and the number of failures are
not at least 15
The estimator of the population proportion based on this plus
X +2
four rule is ^p=
n+ 4
, Plus four estimate ~
p ; mean p and standard deviation
√ p (1− p)/(n+ 4)
Significance test for a single proportion
Significance test = we assume that the null hypothesis is true
To test the hypothesis Ho: p = p0 compute the z statistic
^p− p0
z=
o
√ p0 (1− p 0)
n
H a : p> p 0 is P ( Z ≥ z )
o H a : p< p 0 is P ( Z ≤ z )
o H a : p ≠ p0 is 2 P ( Z ≥|z|)
we do not often use significance tests for a single proportion,
because it is uncommon to have a situation where there is a
precise p0 that we want to test
Choosing a sample size
because we don’t know the value of ^p until we gather the
data, we must guess a value to use in the calculations – the
guessed value is p*
there are two common ways to get p*
o 1. Use the sample estimate from a pilot study or from
similar studies done before
o use p*=0.5. because the margin of error is largest when
^p =0.5; it is a safe choice no matter what the data
later show
z¿ 2
n=( ) p*(1 – p*)
m
o here z* is the critical value for confidence C, and p* is a
guessed value for the proportion of successes in the
future sample
¿
the sample size required when p* = 0.5 is n= 1 ¿ z*/m)2
4
the value of n obtained by this method is not particularly
sensitive to the choice of p* when p* is fairly close to 0.5
e.g. will the proposed sample size of n = 200 be adequate to
provide Rec Sports with the needed information? To address
this we calculate a 95% confidence interval for various values
of ^p
the margins of error will always be the same for ^p and 1 -
^p (p.479)
p. 478 Example 8.7
