Inference on Proportions

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Inference on Proportions
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What are the steps for performing a confidence
interval?

• Assumptions
• Calculations
• Conclusion

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Assumptions

• SRS of context
• Approximate Normal distribution because
• np 10 n(1-p) 10
• Population is at least 10n

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Formula for Confidence interval
Note For confidence intervals, we DO NOT know p
so we MUST substitute p-hat for p in 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.

Step 1 check assumptions!
Step 2 make calculations
Step 3 conclusion in context
We are 95 confident that the true proportion of
adults who believe in ghosts is between 35 and
41.
7
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?
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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.

9
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?
Use p-hat .5
Divide by 1.96
Square both sides
Round up on sample size
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Hypotheses for proportions

• H0 p value
• Ha p value
• where p is the true proportion of context

Use ,
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Formula for hypothesis test
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A company is willing to renew its advertising
contract with a local radio station only if the
station can prove that more than 20 of the
residents of the city have heard the ad and
recognize the companys product. The radio
station conducts a random sample of 400 people
and finds that 90 have heard the ad and recognize
the product. Is this sufficient evidence
for the company to renew its
contract?
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• Assumptions
• Have an SRS of people
• np 400(.2) 80 n(1-p) 400(.8) 320 -
  Since both are greater than 10, this distribution
  is approximately normal.
• Population of people is at least 4000.

Use the parameter in the null hypothesis to check
assumptions!
H0 p .2 where p is the true proportion of
people who Ha p .2 heard the ad
Use the parameter in the null hypothesis to
calculate standard deviation!
Since the p-value a, I fail to reject the null
hypothesis. There is not sufficient evidence to
suggest that the true proportion of people who
heard the ad is greater than .2.