Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is...
-
Upload
morgan-fox -
Category
Documents
-
view
213 -
download
0
Transcript of Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is...
![Page 1: Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is at least 10n.](https://reader036.fdocuments.in/reader036/viewer/2022082820/56649f345503460f94c52333/html5/thumbnails/1.jpg)
Inference on Proportions
![Page 2: Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is at least 10n.](https://reader036.fdocuments.in/reader036/viewer/2022082820/56649f345503460f94c52333/html5/thumbnails/2.jpg)
Assumptions:
• SRS
• Normal distribution
np > 10 & n(1-p) > 10
• Population is at least 10n
![Page 3: Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is at least 10n.](https://reader036.fdocuments.in/reader036/viewer/2022082820/56649f345503460f94c52333/html5/thumbnails/3.jpg)
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.
![Page 4: Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is at least 10n.](https://reader036.fdocuments.in/reader036/viewer/2022082820/56649f345503460f94c52333/html5/thumbnails/4.jpg)
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.
![Page 5: Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is at least 10n.](https://reader036.fdocuments.in/reader036/viewer/2022082820/56649f345503460f94c52333/html5/thumbnails/5.jpg)
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
![Page 6: Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is at least 10n.](https://reader036.fdocuments.in/reader036/viewer/2022082820/56649f345503460f94c52333/html5/thumbnails/6.jpg)
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
*
![Page 7: Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is at least 10n.](https://reader036.fdocuments.in/reader036/viewer/2022082820/56649f345503460f94c52333/html5/thumbnails/7.jpg)
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.
![Page 8: Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is at least 10n.](https://reader036.fdocuments.in/reader036/viewer/2022082820/56649f345503460f94c52333/html5/thumbnails/8.jpg)
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
![Page 9: Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is at least 10n.](https://reader036.fdocuments.in/reader036/viewer/2022082820/56649f345503460f94c52333/html5/thumbnails/9.jpg)
Stop & do homework!
![Page 10: Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is at least 10n.](https://reader036.fdocuments.in/reader036/viewer/2022082820/56649f345503460f94c52333/html5/thumbnails/10.jpg)
Hypotheses for proportions:
H0: p = value
Ha: p > value
where p is the true proportion of context
Use >, <, or ≠
![Page 11: Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is at least 10n.](https://reader036.fdocuments.in/reader036/viewer/2022082820/56649f345503460f94c52333/html5/thumbnails/11.jpg)
Formula for hypothesis test:
statistic of SD
parameter - statisticstatisticTest
z npp
pp
1
ˆ
![Page 12: Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is at least 10n.](https://reader036.fdocuments.in/reader036/viewer/2022082820/56649f345503460f94c52333/html5/thumbnails/12.jpg)
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 company’s 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?
![Page 13: Inference on Proportions. Assumptions: SRS Normal distribution np > 10 & n(1-p) > 10 Population is at least 10n.](https://reader036.fdocuments.in/reader036/viewer/2022082820/56649f345503460f94c52333/html5/thumbnails/13.jpg)
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.
H0: p = .2 where p is the true proportion of people who
Ha: p > .2 heard the ad
05.α1056.25.1
400)8(.2.
2.225.
valuepz
Since the p-value >, 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.
Use the parameter in the null hypothesis to check assumptions!
Use the parameter in the null hypothesis to calculate standard
deviation!