1. Statistical Estimation
Point Estimate – Use a Single Value from a Sample to Estimate an
Unknown Parameter of a Population Є e
Є ≈ e
Unbiased Estimator – e is an Unbiased Estimator of Parameter Є,
If E(e) = Є
E(X)=µ , X is a unbiased estimator of µ , Pop Mean
E(s2) = σ2 , s2 is an unbiased estimator of σ2 , Pop Variance
, s is a slightly biased estimator of σ, Pop Std Dev

2. Interval Estimate – Bound the Parameter between a Upper and Lower Limit
Lower Limit ≤ Parameter ≤ Upper Limit
X - e ≤ µ ≤ X e
e – Error Bound – Maximum difference between X and µ which we expect
to occur.
α – Risk – Probability that the difference between X and µ will exceed e.
We know from the Central Limit Theorem that the distribution for X will be
Normal if we have a large enough sample size.

3. Sample Mean Distribution
α/2
α/2
µ - e
µ + e
Standard Normal Distribution
α/2
α/2
-Zα/2
Zα/2

4. Interval Estimate - X – e ≤ µ ≤ X + e where
and n ≥ 30 so σ ≈ s
Ex: n = 36, X = 20.6 yr and s = 3 yr with α = .05
Confidence Level – C = ( 1 – α )

5. Ex: n = 64, X = 42,000 miles, s = 10,000 miles with C = .90 & α = .10
Sample Size Calculation – Use sample size to control the error

6. Ex: Sample Size to Determine Ave Age for e = .5yr, σ = 3 yr, C = .95

7. Ex: Sample Size for Ave Cost at WSU for e = $250, σ = ?, C =.95

8. Small Sample Size – n < 30
t- Distribution W.S. Gosset -3 -2 -1 1 2 3
Standard Normal
t (d.f. = 25)
t (d.f. = 1)
t (d.f. = 5)

9. Table of Critical Values of tdf
t0.100
t0.050
t0.025
t0.010
t0.005 1
3.078
6.314
12.706
31.821
63.656 2
1.886
2.920
4.303
6.965
9.925 3
1.638
2.353
3.182
4.541
5.841 4
1.533
2.132
2.776
3.747
4.604 5
1.476
2.015
2.571
3.365
4.032 23
1.319
1.714
2.069
2.500
2.807 24
1.318
1.711
2.064
2.492
2.797 25
1.316
1.708
2.060
2.485
2.787 29
1.311
1.699
2.045
2.462
2.756 30
1.310
1.697
2.042
2.457
2.750 40
1.303
1.684
2.021
2.423
2.704 60
1.296
1.671
2.000
2.390
2.660
120
1.289
1.658
1.980
2.358
2.617
1.282
1.645
1.960
2.327
2.576  t  
With df = 24 and a = 0.05, ta = 21

10. Confidence Interval – X - e ≤ µ ≤ X + e
Ex: n =16, X = 20.6 yr, s = 3 yr, C = .95
Sample Size Calculation for Small Sample Problem?

11. Estimation of the Population Proportion
n = sample size
x = number with attribute
= x/n Sample Proportion
Estimator Unbiased?
Standard Error for Estimator?

12. Confidence Interval for the Population Proportion

13. Ex: Percentage who plan to buy a new Car; n = 250, x = 80, C = 95
Ex: Percentage of Voters who will vote for Candidate; n = 400, x = 186, C = .90

14. Sample Size Calculation -
Ex: Voter poll e = .01, C= .95, p = .50

15. Distribution of the Standard Error for Sample Proportion
Ex: Sample Size to Estimate the Proportion Left-Handed

16. Finite Population Correction
Ex: N = 5,000, e = .03, p = .30, C = .95