introduction to confidence intervals using population parameters
DESCRIPTION
Introduction to Confidence Intervals using Population Parameters. Chapter 10.1 & 10.3. Rate your confidence 0 (no confidence) – 100 (very confident). Name my age within 10 years? within 5 years? within 1 year ? What happens to your confidence as the interval (age range) gets smaller?. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/1.jpg)
Introduction to Confidence Intervals using
Population Parameters
Chapter 10.1 & 10.3
![Page 2: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/2.jpg)
Rate your confidence0 (no confidence) – 100 (very confident)
• Name my age within 10 years?• within 5 years?• within 1 year?
• What happens to your confidence as the interval (age range) gets smaller?
![Page 3: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/3.jpg)
Would you agree?
As my age interval decreases your confidence decreases. On the other hand, your confidence increases as the interval widens, because you are given a greater margin of error.
![Page 4: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/4.jpg)
Point Estimate• When we use a single statistic
based on sample data to estimate a population parameter
• Simplest approach• But not always very precise due to
variation in the sampling distribution
![Page 5: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/5.jpg)
Confidence intervals
• Are used to estimate the unknown population parameter
• Formula:
estimate + margin of error
![Page 6: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/6.jpg)
Margin of error• Shows how accurate we believe our
estimate is• The smaller the margin of error, m, the
more precise our estimate of the true parameter.
• Formula:
statistic theofdeviation standard
value
criticalm
![Page 7: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/7.jpg)
Confidence level• Is the success rate of the method
used to construct the interval.
• Using this method, ____% of the intervals constructed will contain the true population parameter.
![Page 8: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/8.jpg)
What does it mean to be 95% confident?
• 95% chance that the true p is contained in the confidence interval
• The probability that the interval contains the true p is 95%
• The method used to construct the interval will produce intervals that contain the true p 95% of the time.
![Page 9: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/9.jpg)
• Found from the confidence level• The upper z-score with probability p lying to its right
under the standard normal curve
Confidence level(C) each tail area z*90% .10/2 =.05 1.64595% .05/2 =.025 1.9699% .01/2 =.005 2.576
• z* can be looked up in table or, by using 2nd VARS #3 invNorm(1.C/2 = Example: 2nd VARS #3 invNorm(1.95/2 = 2.575829
Critical value (z*)
![Page 10: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/10.jpg)
ˆ ˆ(1 )ˆ * p pp zn
Confidence interval for a population proportion:
![Page 11: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/11.jpg)
Steps for doing a confidence interval:
1) State the parameter of interest.
2) Name inference procedure & state assumptions. See assumptions for CI for population parameter on next slide.
3) Calculate the confidence interval using formula.
4) Write a statement about the interval in the context of the
problem.
![Page 12: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/12.jpg)
CI assumptions for a pop. parameterStep 2: Name inference procedure and state assumptions:
1) SRS from population
2) Normality: The number of success and failures are both at least 10. • Note: On AP Test you must show the calculation
below, simply stating the number of successes and failures are both at least 10 isn’t enough.
• np > 10 & n(1-p) > 10.
3) Independence: Population size > 10n
![Page 13: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/13.jpg)
Statement: (memorize!!)
We are ________% confident that the true proportion context lies within the interval ______ and ______.
![Page 14: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/14.jpg)
Assumptions:• The voters were sampled randomly.• 330(.436)=144 & 330(.564)=186, both ≥ 10• Population of eligible voters must be at least 3300 =
10(330).
We are 95% confident that the true proportion of voters that will vote “yes” is between .382 and .490.
Your local newspaper polls a random sample of 330 voters, finding 144 who say they will vote “yes” on the upcoming school budget. Create a 95 % confidence interval for actual sentiment of all voters. 1st Calculate p-hat = 144/330 = .436
.436 1.96.436(.564)
330
.382,.490
![Page 15: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/15.jpg)
Assumptions:• The subjects were sampled randomly• 53 (.27)=14 and 53(.73)=39, both ≥10• The population of subjects using this new medicine must be
at least 530 = 10(53)
We are 95% confident that the true proportion of people that will improve after using the new medication is between .15 and .39.
An experiment finds that 27% of 53 randomly sampled subjects report improvement after using a new medicine. Create a 95% confidence interval for the actual cure rate.
..27 1.96.27(.73)
53
.15,.39
![Page 16: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/16.jpg)
We are 90% confident that the true proportion of people that will improve after using the new medication is between .17 and .37.
90% confidence interval?
.27 1.645.27(.73)
53
.17,.37
![Page 17: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/17.jpg)
How can you make the margin of error smaller?• z* smaller
(lower confidence level)
• s smaller(less variation in the population)
• n larger(to cut the margin of error in half, n, the
sample size must be 4 times as big)
In real life, you can’t adjust
![Page 18: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/18.jpg)
Find a sample size:
• If a certain margin of error is wanted, then to find the sample size necessary for that margin of error use:
Always round up to the nearest person/object!
ˆ ˆ(1 )* p pm zn
![Page 19: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/19.jpg)
Find the sample size required for ±5%, with 98% confidence. Consider the formula for margin of error. We believe the improvement rate to be .27 from our preliminary study.
.05 2.326.27(.73)n
.052 2.3262 .27(.73)n
n 2.3262 (.27)(.73)
.052 426.546
We need to run an experiment with at least 427 people receiving the new medication in order to have a margin of error of ±5%, with 98% confidence.
ˆ ˆ(1 )* p pm zn
![Page 20: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/20.jpg)
When they don’t give you a % of confidence or p-hat:• Use 95% confidence and .5 for p-hat
![Page 21: Introduction to Confidence Intervals using Population Parameters](https://reader035.vdocuments.net/reader035/viewer/2022062323/56816150550346895dd0d998/html5/thumbnails/21.jpg)
What sample size does it take to estimate the outcome for an election with a margin of error of 3%?
.03 1.96.5(.5)n
.032 1.962 .5(.5)n
n 1.962 (.5)(.5)
.032 1068
We need to have a sample size of at least 1068 people to estimate the outcome for an election in order to have a margin of error of ±3%, with 95% confidence.
ˆ ˆ(1 )* p pm zn