1. STA 291 Spring 2008
Lecture 13
Dustin Lueker

2. Statistical Inference: Estimation
Inferential statistical methods provide predictions about characteristics of a population, based on information in a sample from that population
Quantitative variables
Usually estimate the population mean
Mean household income
Qualitative variables
Usually estimate population proportions
Proportion of people voting for candidate A
STA 291 Spring 2008 Lecture 13

3. Two Types of Estimators
Point Estimate
A single number that is the best guess for the parameter
Sample mean is usually at good guess for the population mean
Interval Estimate
Point estimator with error bound
A range of numbers around the point estimate
Gives an idea about the precision of the estimator
The proportion of people voting for A is between 67% and 73%
STA 291 Spring 2008 Lecture 13

4. Point Estimator
A point estimator of a parameter is a sample statistic that predicts the value of that parameter
A good estimator is
Unbiased
Centered around the true parameter
Consistent
Gets closer to the true parameter as the sample size gets larger
Efficient
Has a standard error that is as small as possible (made use of all available information)
STA 291 Spring 2008 Lecture 13

5. Unbiased
An estimator is unbiased if its sampling distribution is centered around the true parameter
For example, we know that the mean of the sampling distribution of equals μ, which is the true population mean
So, is an unbiased estimator of μ
Note: For any particular sample, the sample mean may be smaller or greater than the population mean
Unbiased means that there is no systematic underestimation or overestimation
STA 291 Spring 2008 Lecture 13

6. Biased
A biased estimator systematically underestimates or overestimates the population parameter
In the definition of sample variance and sample standard deviation uses n-1 instead of n, because this makes the estimator unbiased
With n in the denominator, it would systematically underestimate the variance
STA 291 Spring 2008 Lecture 13

7. Efficient
An estimator is efficient if its standard error is small compared to other estimators
Such an estimator has high precision
A good estimator has small standard error and small bias (or no bias at all)
The following pictures represent different estimators with different bias and efficiency
Assume that the true population parameter is the point (0,0) in the middle of the picture
STA 291 Spring 2008 Lecture 13

8. Bias and Efficient
Note that even an unbiased and efficient estimator does not always hit exactly the population parameter.
But in the long run,
it is the best estimator.
STA 291 Spring 2008 Lecture 13

9. Confidence Interval
Inferential statement about a parameter should always provide the accuracy of the estimate
How close is the estimate likely to fall to the true parameter value?
Within 1 unit? 2 units? 10 units?
This can be determined using the sampling distribution of the estimator/sample statistic
In particular, we need the standard error to make a statement about accuracy of the estimator
STA 291 Spring 2008 Lecture 13

10. Confidence Interval
Range of numbers that is likely to cover (or capture) the true parameter
Probability that the confidence interval captures the true parameter is called the confidence coefficient or more commonly the confidence level
Confidence level is a chosen number close to 1, usually 0.90, 0.95 or 0.99
Level of significance = α = 1 – confidence level
STA 291 Spring 2008 Lecture 13

11. Confidence Interval
To calculate the confidence interval, we use the Central Limit Theorem
Use this formula if σ is known
Also, we need a that is determined by the confidence level
Formula for 100(1-α)% confidence interval for μ
STA 291 Spring 2008 Lecture 13

12. Common Confidence Intervals
Confidence level of 0.90
α=.10
Zα/2=1.645
95% confidence interval
Confidence level of 0.95
α=.05
Zα/2=1.96
99% confidence interval
Confidence level of 0.99
α=.01
Zα/2=2.576
STA 291 Spring 2008 Lecture 13

13. Confidence Intervals
This interval will contain μ with a 100(1-α)% probability
We need to know σ to find a confidence interval
If σ is unknown, we may replace it by s (sample standard deviation) but the value Zα/2 needs adjustment to take into account the possible extra variability introduced by s
A different distribution is used, the t-distribution, we will get to this later
STA 291 Spring 2008 Lecture 13

14. Example
Compute a 95% confidence interval for μ if we know that σ=12 and the sample of size 25 yielded a mean of 7
STA 291 Spring 2008 Lecture 13

15. Interpreting Confidence Intervals
“Probability” means that in the long run 100(1-α)% of the intervals will contain the parameter
If repeated samples were taken and confidence intervals calculated then 100(1-α)% of the intervals will contain the parameter
For one sample, we do not know whether the confidence interval contains the parameter
The 100(1-α)% probability only refers to the method that is being used
STA 291 Spring 2008 Lecture 13

16. Interpreting Confidence Intervals
Incorrect statement
With 95% probability, the population mean will fall in the interval from 3.5 to 5.2
To avoid the misleading word “probability” we say that we are “confident”
We are 95% confident that the true population mean will fall between 3.5 and 5.2
STA 291 Spring 2008 Lecture 13

17. Confidence Intervals
Changing our confidence level will change our confidence interval
Increasing our confidence level will increase the length of the confidence interval
A confidence level of 100% would require a confidence interval of infinite length
Not informative
There is a tradeoff between length and accuracy
Ideally we would like a short interval with high accuracy (high confidence level)
STA 291 Spring 2008 Lecture 13