Summary Statistics II 2.2

BOOK M&M&C
CHAPTER 8: INFERENCE FOR PROPORTIONS
8.1 INFERENCE FOR SINGLE PROPORTION
Proportion:refers to the fraction of the total that possesses a certain attribute
LARGE SAMPLE CONFIDENCE INTERVAL
 Inference about a population proportion p from SRS of size n is based on the sample
proportion ^p= X /n
o X is number of successes
o When n is large, ^p has approximately the Normal distribution as size increases with
mean p and standard deviation S ^p= √ p(1− p)/n
 The level C large-sample confidence interval is ^p ± m
o Recommend using interval for 90%, 95%, and 99% confidence when number of
successes and failures are both at least 10 (so can use normal approximation) and
data produced by random sample
 For large samples, the margin of error for confidence level C is m=z∗SE ^p
o Critical value z* is the value for the standard Normal density curve with area C
between −z* and z*
p^ (1− p^ )
 Standard error of ^p is SE ^p=
√ n

SMALLER SAMPLE SIZE
 Alternative procedures such as the plus four estimate of the population proportion are
X +2
recommended ~ p= (add four imaginary observations, two successes and two failures)
n+ 4
o Mean ~ p
~
p ( 1−~
p)
o Standard deviation
√
( n+4 )
o Use large-sample confidence but with ~ p

SIGNIFICANCE TEST (hypothesis)
p^ − p 0
z=
 Tests of H0: p=p0 are based on the z statistic p 0(1−p 0)
√ n
o P-values calculated from the N(0,1) distribution. Use this procedure when expected
number of successes np0 and expected number of failures n(1−p0), are both at
least 10
o Find P-value by calculating probability of getting z statistic this large or larger in the
direction specified by the alternative hypothesis

CHOOSING SAMPLE SIZE
 The sample size required to obtain a confidence interval of approximate margin of error m
for a proportion is found from n=¿ ¿
1

, o p* is a guessed value for the proportion of successes in the future sample
o z* is the standard Normal critical value for the desired level of confidence
o To ensure that the margin of error of the interval is less than or equal to m if guess
1
p¿=0.5 then n= ¿ ¿
4
 Software can be used to determine the sample sizes for significant tests (power)

8.2 COMPARING TWO PROPORTIONS ( p1∧ p2)
LARGE SAMPLE CONFIDENCE INTERVAL
 The large-sample estimate of the difference in two population proportions is D= ^p 1−^p 2
X1 X2
o ^p1− ^p2 are the sample proportions: ^p1= and ^p2=
n1 n2
o Sampling distribution of ^p1− ^p2 with mean p1− p2 is approximately Normal with
larger sample size
 The large-sample level C confidence interval is D ± m
o Recommend using interval for 90%, 95%, or 99% confidence when number of
successes and failures in both samples are at least 10 and are random samples
 The margin of error for confidence level C is m=z∗SE D
o z* is the value for standard Normal density curve with area C between −z* and z*
^p1 (1−^p1) ^p2 (1−^p 2)
 The standard error of the difference D is SED =
√ n1
+
n2

PLUS-FOUR CONFIDENCE INTERVAL
 For smaller sample sizes, the plus four estimate (sample size at least 5) of the difference in
two population proportions is recommended
 Add two imaginary observations, one success and one failure to each of the two samples
~p (1−~ p1 ) ~
p ( 1−~ p2 )
 Confidence interval (~
p1−~
√
p2 ¿ ± z ¿ 1
n1 + 2
+ 2
n2 + 2

SIGNIFICANCE TEST (hypothesis)
^p1−^p2
 Significance tests of comparing two proportions (H0: p1=p2) use the z statistic z=
SE Dp
X1+ X2
o The pooled estimate of the common value of p1 and p2 is ^p=
n1 +n 2
1 1
√
o The pooled standard error is SEDp = ^p (1−^p )
( +
n1 n2 )
o Use this test when the number of successes and the number of failures in each of
the samples are at least 5 + SRS + population at least 10 times as large as samples
1
 Sample size for desired margin of error is given by n=
2 ()
¿ ¿ if p∗¿1 ¿ and p∗¿2 ¿ = 0.5

^p1
 Relative risk is the ratio of two sample proportions RR= for SPSS
^p2
2