Summary
Summary Statistics II 2.2
- Course
- Institution
Summary of the books for course 2.2 statistics
[Show more]
1 review
By: 618533il • 1 year ago
Seller
Follow
dabad23
Reviews received
Content preview
BOOK M&M&CCHAPTER 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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller dabad23. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $11.39. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
77254 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now