1. From the Data at Hand to the World at Large
Part V
From the Data at Hand to the World at Large

2. Chapter 18 Sampling Distribution Models
Modeling the Distribution of sample proportions
From 1000 randomly selected voters (2004)
Poll 1 : John Kerry 49%
Poll 2 : John Kerry 45.9%

3. Assumptions and Conditions
The sampled values must be independent of each other
The sample size, n, must be large.
Conditions
10% Condition
If drawing without replacement then the sample n must be no larger than 10% of the population
Success / Failure condition
The sample size has to be big enough so that both np and nq are greater than 10

4. The Sampling Distribution Model for a Proportion
Provided that the sampled values are independent and the sample size is large enough, the sampling distribution of p is modeled by a normal model with mean
And standard deviation

5. Models for proportions
Exercise 10 page 424

6. Means: The Fundamental Theorem of Statistics
Central Limit Theorem (CLT)
The sampling distribution of any mean becomes normal as the sample grows (independent observations)
As the sample size “n” increases, the mean of n independent values has a sampling distribution that tends toward a normal model with mean equal to the population mean and standard deviation

7. CLT

8. Assumptions and Conditions
Random Sampling Condition
The values must be sampled randomly
Independence assumption
10% condition
The sample size is less than 10% of the population

9. Exercise
Step-by-step page 418
Ex.36 page 426

10. Standard Error
When we estimate the standard deviation of a sampling distribution using statistics found from the data, the estimates are called standard error:
For a proportion
For the sample mean

11. Don’t confuse the sampling distribution with the distribution of the sample
Take a sample
Look at the distribution on a histogram
Calculate summary statistics
Sampling Distribution
Models an imaginary collection of the values that a statistic, might have taken from all the samples that you didn’t get.
We use the sampling distribution model to make statements about how statistics varies

12. Confidence Intervals for Proportions
Example: Infected Sea fan corals at Las Redes Reef (LRR)

13. Confidence Intervals
68% of the samples will have p^ within 1 SE of p. And 95% of all samples will be within p±2SE
We know that for 95% of random samples p^ will be no more than 2SE away from p.
Now from p^ point of view, there is a 95% chance that p is no more than 2SE away from p^

14. Confidence interval
We are 95% confident that between 42.1% and 61.7% of LRR sea fans are infected.
Margin of Error
Certainty vs. Precision
Estimate ± M.E.
The margin of error for our 95% confidence interval was 2SE
For 99.7% confident 3SE
100% Confident % to 100%
Low Confidence % to 52.0

15. Critical Values z*
The number of standard errors to move away from the mean of the sampling distribution to correspond to the specified level of confidence.
Find z* (critical value) for 98% confidence.
For 95%?

16. Confidence interval (one-proportion z-interval)
The critical value z* depends on the particular confidence interval we specify and
Assumptions
Independence
Conditions
Randomization
10% Condition

17. Exercise
#13 Page 444

18. Chapter 20 Testing Hypothesis about proportions
Example:
Metal Manufacturer
Ingots
20% defective (cracks)
After Changes in the casting process:
400 ingots and only 17% defective
IS this a result of natural sampling variability or there is a reduction in the cracking rate?

19. Hypotheses
We begin by assuming that a hypothesis is true (as a jury trial).
Data consistent with the hypothesis:
Retain Hypothesis
Data inconsistent with the hypothesis:
We ask whether they are unlikely beyond reasonable doubt.
If the results seem consistent with what we would expect from natural sampling variability we will retain the hypothesis. But if the probability of seeing results like our data is really low, we reject the hypothesis.

20. Testing Hypotheses Null Hypothesis H0
Specifies a population model parameter of interest and proposes a value for this parameter
Usually:
No change from traditional value
No effect
No difference
In our example H0:p=0.20
How likely is it to get 0.17 from sample variation?