inference on proportions. assumptions: srs normal distribution np > 10 & n(1-p) > 10...
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Inference on Proportions
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Assumptions:
• SRS
• Normal distribution
np > 10 & n(1-p) > 10
• Population is at least 10n
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Formula for Confidence interval:
statistic of SD valuecritical statisticCI
p̂
npp 1*z
Normal curve
Note: For confidence intervals, we DO NOT know p – so we MUST substitute p-hat for pin both the SD & when checking assumptions.
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A May 2000 Gallup Poll found that 38% of a random sample of 1012 adults said that they believe in ghosts. Find a 95% confidence interval for the true proportion of adults who believe in ghost.
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Assumptions:
•Have an SRS of adults
•np =1012(.38) = 384.56 & n(1-p) = 1012(.62) = 627.44 Since both are greater than 10, the distribution can be approximated by a normal curve
•Population of adults is at least 10,120.
41,.35.1012
)62(.38.96.138.
1*ˆ
npp
zP
We are 95% confident that the true proportion of adults who believe in ghost is between 35% and 41%.
Step 1: check assumptions!
Step 2: make calculations
Step 3: conclusion in context
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Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval?
To find sample size:
However, since we have not yet taken a sample, we do not know a p-hat (or p) to use!
npp
zm1
*
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What p-hat (p) do you use when trying to find the sample size for a given margin of error?
.1(.9) = .09
.2(.8) = .16
.3(.7) = .21
.4(.6) = .24
.5(.5) = .25
By using .5 for p-hat, we are using the worst-case scenario and using the largest SD in our calculations.
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Another Gallop Poll is taken in order to measure the proportion of adults who approve of attempts to clone humans. What sample size is necessary to be within + 0.04 of the true proportion of adults who approve of attempts to clone humans with a 95% Confidence Interval?
60125.600
25.96.104.
5.5.96.104.
5.5.96.104.
1*
2
n
n
n
n
npp
zm
Use p-hat = .5
Divide by 1.96
Square both sides
Round up on sample size