introduction to inference confidence intervals
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Introduction to Inference Confidence Intervals. William P. Wattles, Ph.D. Psychology 302. Provides methods for drawing conclusions about a population from sample data. Statistical Inference. Population (parameter). Sample (statistic). The problem. Sampling Error. - PowerPoint PPT PresentationTRANSCRIPT
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Introduction to InferenceConfidence Intervals
William P. Wattles, Ph.D.Psychology 302
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Statistical Inference
Provides methods for drawing conclusions about a population from sample data.
Sample (statistic)
Population (parameter)
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The problem
Sampling Error
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Sampling error results from chance factors that produce a sample statistic different from the population parameter it
represents.
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Inferential statistics
How well does the sample statistic predict the unknown population parameter?
Population
Sample
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Dealing with sampling error
Confidence intervals Hypothesis testing
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Frequency Distribution
Tells what values a variable can take and how often each value occurs
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Sampling Distribution
Tells what values a statistic can take and how often each value occurs.
All possible samplings of a given size Less variable than a raw score
frequency distribution
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Confidence interval
Point versus interval estimation confidence interval= estimate±margin of
error
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Margin of error example
Imagine catering a function where you expect 120 students.
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Margin of error example
Imagine catering a function where you expect 120 students plus or minus 30
What are the upper and lower limits?
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Margin of error example
Imagine catering a function where you expect 120 students plus or minus 30
What are the upper and lower limits? Minimum (lower limit) 90 Maximum (upper limit) 150
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Obtaining confidence intervals
estimate + or - margin of error
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Upper and Lower limits
Bob estimates that Mary weighs 120 pounds “give or take” ten. Calculate the upper and lower limits of his estimate.
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Upper and Lower limits
Bob estimates that Mary weighs 120 pounds “give or take” ten. Calculate the upper and lower limits of his estimate.
Upper 130 Lower 110
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Upper and Lower limits
Tom is giving a party and tells the caterer that he expects 80 friends plus or minus 20. Determine the upper and lower limits
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Upper and Lower limits
Tom is giving a party and tells the caterer that he expects 80 friends plus or minus 20. Determine the upper and lower limits
Upper 100 Lower 60
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Upper and Lower limits
If something costs $250 plus or minus $25, what is the lower limit, the least you would expect to pay? What is the upper limit or the most you would expect to pay.
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Upper and Lower limits
If something costs $250 plus or minus $25, what is the lower limit, the least you would expect to pay?
Upper $275 Lower $225
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The purpose of a confidence interval is to estimate an unknown parameter and an indication of:1. of how accurate the
estimate is 2. how confident we are that
the result is correct.
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Estimating with confidenceAlthough the sample mean is a unique number for any particular sample, if you pick a different sample, you will probably get a different sample mean.
In fact, you could get many different values for the sample mean, and virtually none of them would actually equal the true population mean, .
x
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But the sample distribution is narrower than
the population distribution, by a factor of √n.
n
Sample means,n subjects
Population, xindividual subjects
x
x
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Confidence intervals tell us two things
1. the interval 2. the level of confidence
– C = the confidence interval– p=probability
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Obtaining confidence intervals
Confidence interval for a population mean
nσzM
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Steps to upper limit
1. The Upper limit equals the Mean + Margin of error
2. Margin of error = Z times the standard error (sigma /sqrt of n)
3. Standard Error = std dev/ square root of n
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Determining critical Z
What is the Z for an 80% confidence interval?
We need a number that cuts off the upper 10% and the lower 10%
Table A look for .90 and .10 Z= -1.28 to cut off lower 10% +1.28 to cut off upper 10%
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Determining Critical values of Z
90% .05 1.645 95% .025 1.96 99% .005 2.576 Critical Values: values that mark off a
specified area under the standard normal curve.
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Homework
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Confidence intervals
Example 14.1 Page 360
Want 95% confidence interval
σ =7.5 Mean= 26.8 n=654
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Confidence intervals
Estimate +-Margin of error
Estimate 26.8 Margin of error .60 Upper limit
– 27.4 Lower Limit
– 26.2
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Obtaining a confidence interval for a sample mean value gives you some idea of how far off you may expect the true population mean to be.
nσzM
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Confidence intervals are extremely important in
statistics, because whenever you report a sample mean,
you need to be able to gauge how precisely it estimates the
population mean.
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Characteristics of confidence intervals
The margin of error gets smaller when: – Z gets smaller. More confidence=larger
interval. (i.e., Only 90% confident versus 95%)– sigma gets smaller. Less population
variation equals less noise and more accurate prediction
– n gets larger.
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Example from cliff notes : Suppose that you want to find out the
average weight of all players on the football team. You are select ten players at random and weigh them.
The mean weight of the sample of players is 198, so that number is your point estimate.
The population standard deviation is σ = 11.50. What is a 90 percent confidence interval for the population weight, if you presume the players' weights are normally distributed?
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90% confidence interval
Area to the right 5% Area between that point and the mean
45% Z value 1.65
905 5
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90% Confidence Interval
Another way to express the confidence interval is as the point estimate plus or minus a margin of error; in this case, it is 198 ± 6 pounds.
192-204
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Confidence Intervals
Student Study Times
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Confidence Intervals
Students (269) asked how many hours do you study on a typical weeknight?– sample mean 137 minutes– study times standard
deviation is 65 minutes– Create a 99% confidence
interval
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Problem 14.30
Mean 137Std dev 65n 269z 2.576std error 3.96312margin of error 10.209lower 126.8upper 147.2
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Sampling Distribution Homework
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Problem 14.54 page 390
Wine odors
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DMS odor threshold
Mean 30.4 Std dev 7 95% conf interval
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Problem 14.27
Mean 30.4Std dev 7n 10z 1.96std error 2.213594margin of error 4.338645lower 26.06
upper 34.74
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Caution page 344
The conditions:– Perfect SRS – Population is normal– We know the
population standard deviation (σ)
These conditions are unrealistic.
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Parametric statistics
Assume raw scores form a normal distribution
Assume the data are interval or ratio scores (measurement data)
Assume raw scores are randomly drawn
Robust refers to accuracy of procedure if one of the assumptions is violated,
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Random error versus bias
The margin of error in a confidence interval covers only random sampling errors.
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The End