Transcript 12.1.1 - GEOCITIES.ws

Inference for Proportions
Section 12.1.1
Starter 12.1.1
• Do dogs who are house pets have higher
cholesterol than dogs who live in a
research clinic? A clinic measured the
cholesterol level in all 23 of its dogs and
found a mean level of 174 with s.d. of 44.
They also measured 26 house pets
brought in to be neutered one week and
found a mean of 193 with s.d. of 68.
• Is this strong evidence that house pets
have higher cholesterol than clinic dogs?
• What is wrong with this study?
Today’s Objectives
• Students will use a formula to form a confidence
interval for a population proportion.
• Students will use a formula to perform a
hypothesis test about a population proportion.
California Standard 17.0
Students determine confidence intervals for a simple
random sample from a normal distribution of data and
determine the sample size required for a desired margin
of error.
The Big Picture (so far)
• Chapter 11: Estimating means
– One population: μ = some constant
– Two populations: μ1 = μ2
• Chapter 12: Estimating proportions
The Distribution of Sample Proportions
• Recall from chapter 9 that if we take many
samples of a population proportion that all
such samples have a normal distribution.
• The mean of the sampling distribution is the
true population proportion.
• The standard deviation of the distribution is
given by the formula
p(1  p)
n
• Since the distribution of sample proportions
is normal, and we know its standard
deviation, we can use z tests, not t tests.
Assumptions
• Data are an SRS from the population
• Population size is at least ten times
sample size
• Sample size is large enough that both the
expected “yes” and “no” counts are 10 or
more:
np  10
n(1  p )  10
(Notice that we use p, not p-hat)
Assumptions So Far…
Procedure
• One population t for means
• Two population t for means
• One population z for proportions
Assumptions
• SRS, Normal Dist
• SRS, Normal, Independent
• SRS, large pop, 10 succ/fail
But we don’t know p. Now what?
Hypothesis Tests
• We start by assuming some value po
• Since we assume po is true, use it in the
standard deviation formula as is: po (1  po )
n
• Now find the z statistic as usual z 
pˆ  po
po (1  po )
n
Hypothesis Test Example
•
A coin is flipped 4040 times and heads
comes up 2048 times. Is this good
evidence that the coin is not fair?
1.
2.
3.
4.
Are the three assumptions met?
State the null and alternative hypotheses
What is p-hat?
Find the z statistic and associated p-value;
draw a conclusion
Assumptions
• The flips are an SRS of all possible flips.
• The population (all possible flips) is more
than 10 times the sample size.
• Assuming p = .5, we expect .5 x 4040 =
2020 heads and 2020 tails. Both are
greater than 10, so assumption is met.
• Ho: p = .5
Ha: p ≠ .5 (Why 2-sided?)
• p-hat is 2048 / 4040 = .5069
Calculations
z
pˆ  po
.5069  .5

 .8771
po (1  po )
(.5)(.5)
4040
n
• normalcdf(.8771,999) = .19
• Because Ha is 2-tailed, p = .19 x 2 = .38
• There is not sufficient evidence (p = .38) to
support a claim that the coin is unfair.
Confidence Intervals
• We don’t know p, so replace standard
deviation with standard error (SE).
• Use the same formula, but replace p with
p-hat:
pˆ (1  pˆ )
SE 
n
• Form a C.I. as usual: estimate ± z*SE
pˆ (1  pˆ )
C.I .  pˆ  z *
n
Confidence Interval Example
• A national AIDS survey found that 170 of
2673 adult heterosexuals had multiple
partners.
• Does this meet the three assumptions
needed for inference?
• Form a 99% C.I. for the true proportion of
adult heterosexuals with multiple partners.
Assumptions
• The actual survey design was a complex
stratified sample. The result was close to an
SRS and may be used.
• The number of adult heterosexuals is much
larger than 10 times the sample size.
• The counts of “yes” and “no” are much larger
than 10:
– 2673 x .0636 = 170
– 2673 x .9364 = 2503
(Or: more than 10 yes)
(Or: more than 10 no)
Calculations
pˆ (1  pˆ )
C.I .  pˆ  z *
n
.0636 x.9364
.0636  2.576
2673
.0636 ± .0122 = (.0514, .0758)
So we are 99% confident that the true
proportion of adult heterosexuals who
have multiple partners is between 5.1%
and 7.6%.
Today’s Objectives
• Students will use a formula to form a confidence
interval for a population proportion.
• Students will use a formula to perform a
hypothesis test about a population proportion.
California Standard 17.0
Students determine confidence intervals for a simple
random sample from a normal distribution of data and
determine the sample size required for a desired margin
of error.
Homework
• Read pages 660 – 668
• Do problems 5-8