1. Chapter 4: Sampling and Statistical Inference
Part 2: Estimation

2. Types of Estimates
Point estimate – a single number used to estimate a population parameter
Interval estimate – a range of values between which a population parameter is believed to be

3. Common Point Estimates

4. Theoretical Issues
Unbiased estimator – one for which the expected value equals the population parameter it is intended to estimate
The sample variance is an unbiased estimator for the population variance

5. Confidence Intervals
Confidence interval (CI) – an interval estimated that specifies the likelihood that the interval contains the true population parameter
Level of confidence (1 – a) – the probability that the CI contains the true population parameter, usually expressed as a percentage (90%, 95%, 99% are most common).

6. Confidence Intervals for the Mean - Rationale

7. Confidence Interval for the Mean – s Known
A 100(1 – a)% CI is: x  z/2(/n)
z/2 may be found from Table A.1 or using the Excel function NORMSINV(1-a/2)

8. Confidence Interval for the Mean, s Unknown
A 100(1 – a)% CI is: x  t/2,n-1(s/n)
t/2,n-1 is the value from a t-distribution with n-1 degrees of freedom, from Table A.3 or the Excel function TINV(a, n-1)

9. Relationship Between Normal Distribution and t-distribution
The t-distribution yields larger confidence intervals for smaller sample sizes.

10. PHStat Tool: Confidence Intervals for the Mean
PHStat menu > Confidence Intervals > Estimate for the mean, sigma known…, or Estimate for the mean, sigma unknown…

11. PHStat Tool: Confidence Intervals for the Mean - Dialog
Enter the confidence level
Choose specification of sample statistics
Check Finite Population Correction box if appropriate

12. PHStat Tool: Confidence Intervals for the Mean - Results

13. Confidence Intervals for Proportions
Sample proportion: p = x/n
x = number in sample having desired characteristic
n = sample size
The sampling distribution of p has mean p and variance p(1 – p)/n
When np and n(1 – p) are at least 5, the sampling distribution of p approach a normal distribution

14. Confidence Intervals for Proportions
A 100(1 – a)% CI is:
PHStat tool is available under Confidence Intervals option

15. Confidence Intervals and Sample Size
CI for the mean, s known
Sample size needed for half-width of at most E is n  (z/2)2(s2)/E2
CI for a proportion
Sample size needed for half-width of at most E is
Use p as an estimate of p or 0.5 for the most conservative estimate

16. PHStat Tool: Sample Size Determination
PHStat menu > Sample Size > Determination for the Mean or Determination for the Proportion
Enter s, E, and confidence level
Check Finite Population Correction box if appropriate

17. Confidence Intervals for Population Total
A 100(1 – a)% CI is:
N x  tn-1,a/2
PHStat tool is available under Confidence Intervals option

18. Confidence Intervals for Differences Between Means
Population 1
Population 2
Mean m1 m2
Standard deviation s1 s2
Point estimate
x1
x2
Sample size n1 n2
Point estimate for the difference in means, m1 – m2, is given by x1 - x2

19. Independent Samples With Unequal Variances
A 100(1 – a)% CI is:
x1 -x2  (ta/2, df*)
Fractional values rounded down
df* =

20. Independent Samples With Equal Variances
A 100(1 – a)% CI is:
x1 -x2  (ta/2, n1 + n2 – 2)
where sp is a common “pooled” standard deviation. Must assume the variances of the two populations are equal.

21. Paired Samples A 100(1 – a)% CI is: D  (tn-1,a/2) sD/n
Di = difference for each pair of observations
D = average of differences
PHStat tool available in the Confidence Intervals menu

22. Differences Between Proportions
A 100(1 – a)% CI is:
Applies when nipi and ni(1 – pi) are greater than 5

23. Sampling Distribution of s
The sample standard deviation, s, is a point estimate for the population standard deviation, s
The sampling distribution of s has a chi-square (c2) distribution with n-1 df
See Table A.4
CHIDIST(x, deg_freedom) returns probability to the right of x
CHIINV(probability, deg_freedom) returns the value of x for a specified right-tail probability

24. Confidence Intervals for the Variance
A 100(1 – a)% CI is:

25. PHStat Tool: Confidence Intervals for Variance - Dialog
PHStat menu > Confidence Intervals > Estimate for the Population Variance
Enter sample size, standard deviation, and confidence level

26. PHStat Tool: Confidence Intervals for Variance - Results

27. Time Series Data
Confidence intervals only make sense for stationary time series data

28. Probability Intervals
A 100(1 – a)% probability interval for a random variable X is an interval [A,B] such that P(A X  B) = 1 – a
Do not confuse a confidence interval with a probability interval; confidence intervals are probability intervals for sampling distributions, not for the distribution of the random variable.