1. Estimation
Goal: Use sample data to make predictions regarding unknown population parameters
Point Estimate - Single value that is best guess of true parameter based on sample
Interval Estimate - Range of values that we can be confident contains the true parameter

2. Point Estimate
Point Estimator - Statistic computed from a sample that predicts the value of the unknown parameter
Unbiased Estimator - A statistic that has a sampling distribution with mean equal to the true parameter
Efficient Estimator - A statistic that has a sampling distribution with smaller standard error than other competing statistics

3. Point Estimators
Sample mean is the most common unbiased estimator for the population mean m
Sample standard deviation is the most common estimator for s (s2 is unbiased for s2)
Sample proportion of individuals with a (nominal) characteristic is estimator for population proportion

4. Confidence Interval for the Mean
Confidence Interval - Range of values computed from sample information that we can be confident contains the true parameter
Confidence Coefficient - The probability that an interval computed from this method on a random sample will contain the true unknown fixed parameter (.90,.95,.99 are typical values)
Central Limit Theorem - Sampling distributions of sample mean is approximately normal in large samples

5. Confidence Interval for the Mean
In large samples, the sample mean is approximately normal with mean m and standard error
Thus, we have the following probability statement:
That is, we can be very confident that the sample mean lies within 1.96 standard errors of the (unknown) population mean

6. Confidence Interval for the Mean
Problem: The standard error is unknown (s is also a parameter). It is estimated by replacing s with its estimate from the sample data:
95% Confidence Interval for m :

7. Confidence Interval for the Mean
Most reported confidence intervals are 95%
By increasing confidence coefficient, width of interval must increase
Rule for (1-a)100% confidence interval:

8. Properties of the CI for a Mean
Confidence level refers to the fraction of time that CI’s would contain the true parameter if many random samples were taken from the same population
The width of a CI increases as the confidence level increases
The width of a CI decreases as the sample size increases
CI provides us a credible set of possible values of m with a small risk of error

9. Confidence Interval for a Proportion
Population Proportion - Fraction of a population that has a particular characteristic (falling in a category)
Sample Proportion - Fraction of a sample that has a particular characteristic (falling in a category)
Sampling distribution of sample proportion (large samples) is approximately normal

10. Confidence Interval for a Proportion
Parameter: p (a value between 0 and 1, not )
Sample - n items sampled, X is the number that possess the characteristic (fall in the category)
Sample Proportion:
Mean of sampling distribution: p
Standard error (actual and estimated):

11. Confidence Interval for a Proportion
Criteria for large samples
0.30 < p <  n > 30
Otherwise, X > 10, n-X > 10
Large Sample (1-a)100% CI for p :

12. Choosing the Sample Size
Bound on error (aka Margin of error) - For a given confidence level (1-a), we can be this confident that the difference between the sample estimate and the population parameter is less than za/2 standard errors in absolute value
Researchers choose sample sizes such that the bound on error is small enough to provide worthwhile inferences

13. Choosing the Sample Size
Step 1 - Determine Parameter of interest (Mean or Proportion)
Step 2 - Select an upper bound for the margin of error (B) and a confidence level (1-a)
Proportions (can be safe and set p=0.5):
Means (need an estimate of s):

14. Small-sample Inference for m
t Distribution:
Population distribution for a variable is normal
Mean m, Standard Deviation s
The t statistic has a sampling distribution that is called the t distribution with (n-1) degrees of freedom:
Symmetric, bell-shaped around 0 (like standard normal, z distribution)
Indexed by “degrees of freedom”, as they increase the distribution approaches z
Have heavier tails (more probability beyond same values) as z
Table B gives tA where P(t > tA) = A for degrees of freedom 1-29 and various A

15. Probability
Cri t ical
Values
Degrees of
Freedom
Critical Values

17. Small-Sample 95% CI for m
Random sample from a normal population distribution:
t.025,n-1 is the critical value leaving an upper tail area of .025 in the t distribution with n-1 degrees of freedom
For n  30, use z.025 = 1.96 as an approximation for t.025,n-1

18. Confidence Interval for Median
Population Median - 50th-percentile (Half the population falls above and below median). Not equal to mean if underlying distribution is not symmetric
Procedure
Sample n items
Order them from smallest to largest
Compute the following interval:
Choose the data values with the ranks corresponding to the lower and upper bounds