1. Political Research & Analysis (PO657) Session V- Normal Distribution, Central Limit Theorem & Confidence Intervals

2. Goals- Lab Session Continuing with Inferential Statistics.
Using Decriptives to get basic info about Interval-level variables
2. Obtain Confidence Intervals for Sample Mean in SPSS
Calculating by hand Confidence Intervals
3. Introduction to T-Test Procedures in SPSS (More of this next week when we cover T-tests extensively.)

3. Course so far…
So far we have observed patterns, created explanations, framed hypotheses at different measurement levels, & analyzed relationships.
Reality of Political Science research is that we must work often with samples rather than with population data.
As students of Political Research, we talk about the confidence we have in our results- Notions of Confidence & Probability.

4. Statistical Terms Recap:
Three concepts that should not be confused: margin of error, standard error and standard deviation.
Margin of error is half the width of a confidence interval.
Standard error measures the variation of a statistic.
Standard deviation measures the variation within a population or sample.

5. When dealing with sample data, we encounter random sampling error.
We need to know whether our results fall within the boundaries of random sampling error?
We ask the fundamental question, how confident we can be that the difference between the two sample means reflects the true differences between women and men in the population? We look at the Standard error- The standard error of the difference between the sample means.

6. Application of Confidence Intervals?
Frequently used by Polling/Market Research Organizations.
Not just used by Academics/Statisticians!
Example: Voting Intention, YouGov UK & YouGov France, TNS Brussels- All these companies use Confidence Intervals to ascertain how confident about the results they are. Often Confidence Intervals/Margin of Error can make or break MR companies.

7. In all seriousness, we can work it out by hand, but easier to get SPSS to do this for us. 
The Formula gives us two pieces of information
Step 1: Work out the Standard Error
Step 2: Work out the Confidence Interval with the information you now have on the Standard Error

8. We get our Z score from the Z Table, i. e. 1
We get our Z score from the Z Table, i.e creates a 95% Critical Region creates a 99% Region.
Construct a 95% Confidence Interval
(1.96) (.052)= 4.66 (Upper Bound)
(1.96) (.052)= 4.46 (Lower Bound)
Construct a 99% Confidence Interval
“Simple Plug & Chug”
4.561+(2.58) (.052)= 4.70
(2.58) (.052)= 4.43
4.561 – 2.58x.052= 4.43 (Use a Calculator or the Maple TA Programme)

9. Descriptives & One-Sample T Test Example
Open up NES2008
Analyse- Descriptive Statistics- Descriptives
Variable- Click spend 10 into the box. (This is a measure of individuals’ opinions about federal spending.)
Click Options Button- We want the Mean, SD, Min, Max Values, also click SE of the Mean
N.B. The standard error of a sample mean is based on the 1. SD & the 2. Size of the sample
SPSS interprets the Standard Error by dividing the SD by the SQ Root of the sample size.

10. SPSS Output:
Mean Value of Spend 10 is 4.56 SD= 2.50 (Rounded up) We want to know how far off the mark our sample estimate of 4.56 is. SPSS calculates the Standard Error by dividing the SD by the Square Root of the sample size. Example: 2.50/sqrt (2323) = 2.50/48.20 = tells us how much the sample mean of 4.56 departs by chance from the population mean. Standard Error allows us to make inferences about the population mean.

11. Steps to remember:
Use One-Sample T Test to obtain 95% Confidence Interval of Spend10.
Click Analyse---> Compare Means----> One Sample T Test.
Set Test Value to 0 (Click Options & set the Confidence Interval % to 95%.) Then run the analysis.
One-Sample Statistics Table & One-Sample Test will come up.

12. Interpretation:
When analysing Confidence Intervals, we want the Lower & Upper 95% boundaries for the Confidence Intervals of the difference.
N.B. 95% Confidence Interval is probabilistic rather than deterministic.
How much confidence do we invest in our sample of 4.56?
Is the true population mean around 4.56?

13. There is a 95% probability that the population mean lies between 4
There is a 95% probability that the population mean lies between 4.46 at the low end and 4.66 at the high end.
In other words, 5% of the time we would get a sample mean that falls outside the boundaries of 4.46 and 4.66 range.
So, there is a 2.5% chance that the population mean is below 4.46 and a chance that the population mean is greater than 4.66.
Why is this? Because the curve is symmetrical, one-half of the remaining 5 % (2.5%) will fall in the negative tail below 4.46 and one-half, the other 2.5% will fall in the positive tail, above Page 126 in the SPSS Companion graphically shows this phenomenon.

14. We can also test the 90% Confidence Interval of our Mean. (N. B
We can also test the 90% Confidence Interval of our Mean. (N.B. 95% Confidence Interval is the best one to use.)
We can be confident (90%) that the population of spend10 lies between at the low end at 4.65 at the high end
Probability of .10 that the population lies between these limits, probability that population mean is < than 4.48 and 0.5 probability >4.65

15. Normal Distribution Curve (CLT)
3 things about the Normal distribution:
Symmetric: Mean = Median = Mode
Total area under the curve is 1
Fixed Proportions

16. “Standard Normal” Distribution
The % Rule
68% of the observations fall within 1 standard deviation of the mean
95% of the observations fall within 2 (actually, 1.96) standard deviations of the mean
99.7% of the observations fall within 3 standard deviations of the mean
So, for a normal distribution, almost all values lie within 3 standard deviations of the mean
Remember that the rule applies to ALL normal distributions …and ONLY to normal distributions

17. Testing a Hypothetical claim about a Sample Mean:
We might want to test the hypothesis that PolSci students are more likely to favour government spending than most people.
We test this idea through asking PolSci students a series of 10 questions about federal programs- the same set of 10 questions that appear in NES 2008.
NES reports a mean of 4.56, however our study might report a Mean of 4.63 for the PolSci students.

18. Interpretation:
However, this difference is not statistically significant at the .05 level- This is because 4.63 does not exceed the NES Sample’s Upper Confidence boundary of 4.65
We have to say this because the PolSci students’ Mean of 4.63 does not exceed the NES Sample’s upper confidence boundary of 4.65.
We need to look at the P-value association.
- Click Analyse----> Compare Means--> One Sample T test. This time click into the Test Box.

19. SPSS calculates 4. 56-4. 63 and reports the result -
SPSS calculates and reports the result -.07 in the Mean Difference Column.
SPSS reports the probability of obtaining a difference of .07 in either direction: a difference of -.07 or a difference of +.07.
Spend10’s Mean falls 1.34 Standard Errors below the Political Science Major’s Mean. Sig 2-Tailed Statistic shows the curve that lies below t= and above t= N.B. We are testing the hypothesis that the majors’ mean is greater than the population mean.t

20. Independent- Samples T Test (Extension)
Hypothesis Testing- N.B. We always begin by assuming the Null Hypothesis to be correct.
H1- In a comparison of Individuals, men will give gays lower feeling thermometer ratings than women.
H2- In a comparison of individuals, men will place more importance on international policy goals than women.
N.B. Remember we test our Hypothesis against the Null Hypothesis.
- Click Analyse--Compare Means----> Independent- Samples T Test

21. Click gay_therm & global_goals into the Test Variable (s) Panel
Click Options
Change the Confidence Interval to 90. Click continue
Click Gender into the Grouping Variable Box. Click Define Groups.
In the Define Groups Window, type “1” in the Group 1 Box and type “2” in the Group 2 Box. Click continue