1. Statistics: Unlocking the Power of Data Lock 5 STAT 250 Dr. Kari Lock Morgan Estimation: Confidence Intervals SECTION 3.2 Confidence Intervals (3.2)

2. Statistics: Unlocking the Power of Data Lock 5 Question of the Day Do uncommitted members of a group (of fish) make it more or less democratic?

3. Statistics: Unlocking the Power of Data Lock 5 Animal Behavior: Fish Democracies Golden shiners are small freshwater fish with a strong tendency to stick together in schools Golden shiners were each trained to prefer a particular color, then released as a pack Which color does the pack go to? Couzin, I. et. al. (2011). “Uninformed Individuals Promote Democratic Consensus in Animal Groups,” Science, 344(6062), 1578-1580.

4. Statistics: Unlocking the Power of Data Lock 5 The Golden Shiner Experiment Trained to swim to a particular color (yellow or blue) with treats

5. Statistics: Unlocking the Power of Data Lock 5 Question #1 What parameter are we estimating? How long does it take Golden shiners to swim to the yellow target?

6. Statistics: Unlocking the Power of Data Lock 5 Question #1 How long does it take Golden shiners to swim to the yellow target? Population parameter: Sample statistic: What else do we want to know?

7. Statistics: Unlocking the Power of Data Lock 5 Interval Estimate Our interval will be centered around our best guess, the sample statistic of 51 seconds How wide should the interval be? How can we use the standard error of 2.4 seconds to determine this? To answer this, we need a formal definition for a confidence interval…

8. Statistics: Unlocking the Power of Data Lock 5 Confidence Interval A confidence interval for a parameter is an interval computed from sample data by a method that will capture the parameter for a specified proportion of all samples The success rate (proportion of all samples whose intervals contain the parameter) is known as the confidence level A 95% confidence interval will contain the true parameter for 95% of all samples

9. Statistics: Unlocking the Power of Data Lock 5 Confidence Intervals www.lock5stat.com/StatKey The parameter is fixed The statistic is random (depends on the sample) The interval is random (depends on the statistic)

10. Statistics: Unlocking the Power of Data Lock 5 95% of 95% confidence intervals will contain the true parameter value

11. Statistics: Unlocking the Power of Data Lock 5 Confidence Level Suppose you each go out, collect a good random sample of data, compute the sample statistic, and correctly create a 95% confidence interval. What percentage of you will have intervals that miss the truth? a) 100% b) 0% c) 95% d) 5%

12. Statistics: Unlocking the Power of Data Lock 5 Margin of Error One common form for an interval estimate is statistic ± margin of error where the margin of error reflects the precision of the sample statistic as a point estimate for the parameter.

13. Statistics: Unlocking the Power of Data Lock 5 Margin of Error The higher the standard deviation of the sampling distribution, the the margin of error. a) higher b) lower

14. Statistics: Unlocking the Power of Data Lock 5 95% of statistics will be within 2SE of the true parameter value (95% rule) truth 95% of statistics 2 SE

15. Statistics: Unlocking the Power of Data Lock 5 The interval statistic ± 2SE will include the parameter 95% of the time truth 2 SE Statistic ± 2 SE will capture the truth for the statistics colored black (middle 95%) but not red (extreme 5%)

16. Statistics: Unlocking the Power of Data Lock 5 95% Confidence Interval If the sampling distribution is relatively symmetric and bell-shaped, a 95% confidence interval can be estimated using statistic ± 2 × SE

17. Statistics: Unlocking the Power of Data Lock 5 95% Confidence Interval

18. Statistics: Unlocking the Power of Data Lock 5 How Long to Yellow?

19. Statistics: Unlocking the Power of Data Lock 5 Interpreting a Confidence Interval 95% of all samples yield intervals that contain the true parameter We say we are “95% sure” or “95% confident” that one interval contains the truth. Interpretations of a confidence interval should always include the context of the problem We are 95% confident that it takes between 46.2 and 55.8 seconds, on average, for a fish trained in this fashion to swim to the yellow target.

20. Statistics: Unlocking the Power of Data Lock 5 The Golden Shiner Experiment Golden shiners have natural preference for yellow, so those trained to yellow had stronger opinions/preferences Packs were formed with a minority yellow- trained (strong opinions) and a majority blue-trained (weak-opinions) The pack swims together towards one color Which color?

21. Statistics: Unlocking the Power of Data Lock 5 Question #2 What parameter are we estimating? Does a minority with a stronger opinion or a majority with a weaker opinion win?

22. Statistics: Unlocking the Power of Data Lock 5 Question #2 Population parameter: Sample statistic: Standard error: SE = 0.04 Give and interpret a 95% confidence interval. Does a minority with a stronger opinion or a majority with a weaker opinion win?

23. Statistics: Unlocking the Power of Data Lock 5 Does Minority or Majority Win?

24. Statistics: Unlocking the Power of Data Lock 5 Indifferent Fish a) The strong-opinioned minority b) The weak-opinioned majority Experiment was conducted the same as before, but this time they added 10 indifferent fish to each group What’s the effect of these indifferent fish? Do they benefit the strong-opinioned minority or the weak-opinioned majority? I think they will benefit…

25. Statistics: Unlocking the Power of Data Lock 5 Question #3 What parameter are we estimating? What is the effect of indifferent fish on the proportion of times the majority wins?

26. Statistics: Unlocking the Power of Data Lock 5 Question #3 What is the effect of indifferent fish on the proportion of times the majority wins? Parameter: Statistic: Give and interpret a 95% confidence interval.

27. Statistics: Unlocking the Power of Data Lock 5 What’s the Effect of Indifferent Fish?

28. Statistics: Unlocking the Power of Data Lock 5 Common Misinterpretations Misinterpretation 1: “A 95% confidence interval contains 95% of the data in the population” Misinterpretation 2: “I am 95% sure that the mean of a sample will fall within a 95% confidence interval for the mean” Misinterpretation 3: “95% of all sample means will fall within this 95% confidence interval” Misinterpretation 4: “The probability that the population parameter is in this particular 95% confidence interval is 0.95”

29. Statistics: Unlocking the Power of Data Lock 5 Confidence Intervals 1) What parameter are you estimating? 2) What is the relevant sample statistic? 3) What is the standard error of this statistic? 4) Calculate a 95% interval with statistic 2 SE. 5) Interpret in context.

30. Statistics: Unlocking the Power of Data Lock 5 Confidence Intervals Population Sample... Calculate statistic for each sample Sampling Distribution Standard Error (SE): standard deviation of sampling distribution Margin of Error (ME) (95% CI: ME = 2×SE) statistic ± ME

31. Statistics: Unlocking the Power of Data Lock 5 Summary To create a plausible range of values for a parameter: o Take many random samples from the population, and compute the sample statistic for each sample o Compute the standard error as the standard deviation of all these statistics o Use statistic  2  SE One small problem…

32. Statistics: Unlocking the Power of Data Lock 5 To Do Read Section 3.2 Do HW 3.1, 3.2 (due Friday, 10/2)