1. Inferential Statistics 10.1 Confidence Intervals
CHAPTER 10
Inferential Statistics
10.1
Confidence Intervals

2. Statistical Inference
Statistical Inference provides methods for drawing conclusions about a population from sample data.
Consider what the population mean must be, based on
Consider what the population proportion must be, based on .

3. Draft Lotteries & Drug Studies
With an r = – was there something “funny going on” – unfair – with the Vietnam Draft? Well … P < 0.001
What if 40 patients were split into two groups to get a drug. 20 get the real thing; 20 get the Placebo. 12 feel better in the real group; 8 in the Placebo. So? Turns out P is about 0.2 … not too rare.

4. 1969 Vietnam Draft Lottery r = -0.226
Subtle, yet significant! … not really perceptible!
r =

5. Consequences
It was quickly noticed that draft ranks were not uniformly distributed over the year. In particular, birthdays in December had lower (earlier) draft ranks, on average, than birthdays in other months. This led to complaints that the lottery was not random as the legislation required. Analysis of the lottery method suggested that the procedure (mixing the capsules in the shoe box and dumping them into the jar) did not mix the capsules sufficiently, however the-less-than-random lottery was allowed to stand. The lottery system was later modified to produce a more random result.

6. SAT MATH in CALIFORNIA N = 350,000 … n = 500 …
The margin of error for a 95% CI:
“We are 95% confident that this interval contains the true population mean score for all California Seniors”

7. Samples and Confidence Intervals

8. Samples and Confidence Intervals

9. Samples and Confidence Intervals

10. Conditions
It is appropriate to construct a Confidence Interval only when:
The data were chosen randomly from the population of interest
The sampling distribution is approximately normal
The individuals in the sample were independently chosen
The Population is at least ten times as big as the sample

11. From z to z*
The z value is something we have calculated with known values of x, mu, and sigma.
NOW – we know what we want to set out area at … our Confidence Level dictates that.
z* is just the perfect value of z to create a desired level of confidence.

12. z* of 1.28  CL = 80%

13. z* of whatever  CL = C% Total tail probability =  (alpha) = 1 -C

14. Typical z-values 90% CL  tail area = .05  z* = 1.645

15. CRITICAL VALUES

16. CONFIDENCE INTERVAL FOR A POPULATION MEAN

17. VIDEO SCREEN TENSION Construct a 90% Confidence Interval 269.5 297.0
269.6
283.3
304.8
280.4
233.5
257.4
317.5
327.4
264.7
307.7
310.0
343.3
328.1
342.6
338.8
340.1
374.6
336.1

18. BIG n versus SMALL n

19. INFERENCE TOOLBOX
Step 1: Identify the population and parameter to be inferred
Step 2: Chose the appropriate Inference Procedure – Verify Conditions
Step 3: CI = estimate ± margin of error
Step 4: Interpret

20. 10.6: SURVEYING HOTEL MANAGERS

21. COMPARATIVE EFFECTS OF z, , and n
Making z smaller … Bigger?
Making  smaller … Bigger?
Making n smaller … Bigger?
By what factor do we need to increase the sample size by to reduce the margin of error by one-half?

22. VIDEO SCREEN TENSION (revisited)
Make a 99% CI now instead

23. CHOOSING A SAMPLE SIZE
The company that we are still working with wants to report the screen tension within ±5 mV with 95% confidence.
What sample size will achieve this?

24. CAUTIONS Careful with RANDOM SRSs Only SRSs – no stratified
Watch bias in data collection – cannot be mathematically “corrected”
Watch out for outliers – eliminate?
Watch for skewing – or any non-Normality – as the true C-level will not be accurate
YOU MUST KNOW .
Confidence is not Probability