STAT 250 Dr Kari Lock Morgan Estimation Confidence

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STAT 250 Dr. Kari Lock Morgan Estimation: Confidence Intervals SECTION 3. 2 • Confidence

STAT 250 Dr. Kari Lock Morgan Estimation: Confidence Intervals SECTION 3. 2 • Confidence Intervals (3. 2) Statistics: Unlocking the Power of Data Lock 5

Margin of Error One common form for an interval estimate is statistic ± margin

Margin of Error One common form for an interval estimate is statistic ± margin of error where the margin of error reflects the uncertainty of the sample statistic. Statistics: Unlocking the Power of Data Lock 5

http: //www. politico. com/story/2017/02/poll-majorities-oppose-trump-immigration-actions-234750 Statistics: Unlocking the Power of Data Lock 5

http: //www. politico. com/story/2017/02/poll-majorities-oppose-trump-immigration-actions-234750 Statistics: Unlocking the Power of Data Lock 5

Interval Estimate 51% oppose; margin of error ≅ 3% Give an interval estimate for

Interval Estimate 51% oppose; margin of error ≅ 3% Give an interval estimate for the true proportion of voters that oppose Trump’s executive order: a) (50%, 52%) b) (48%, 54%) c) (47%, 55%) d) (46%, 56%) e) (45%, 57%) Statistics: Unlocking the Power of Data Lock 5

Interval Estimate Is the headline for the article justified? a) Yes b) No Statistics:

Interval Estimate Is the headline for the article justified? a) Yes b) No Statistics: Unlocking the Power of Data Lock 5

Question of the Day Do uncommitted members of a group (of fish) make it

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

Fish Democracies �Golden shiners are small freshwater fish with a strong tendency to stick

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

Question #1 How long does it take Golden shiners to swim to the yellow

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

Question #1 How long does it take Golden shiners to swim to the yellow

Question #1 How long does it take Golden shiners to swim to the yellow target? � Statistics: Unlocking the Power of Data Lock 5

Distance from parameter to statistic gives distance from statistic to parameter p SE can

Distance from parameter to statistic gives distance from statistic to parameter p SE can be used to determine width of interval! Rare for statistics to be further than this from parameter ? So rare for parameter to be further than this from statistic Statistics: Unlocking the Power of Data Lock 5

95% Confidence Intervals A 95% confidence interval will contain the true parameter value for

95% Confidence Intervals A 95% confidence interval will contain the true parameter value for 95% of all samples. � 95% is known as the confidence level. (Other confidence levels to come next class. ) Statistics: Unlocking the Power of Data Lock 5

Confidence Intervals �www. lock 5 stat. com/Stat. Key � The parameter is fixed �The

Confidence Intervals �www. lock 5 stat. com/Stat. Key � The parameter is fixed �The statistic is random (depends on the sample) �The interval is random (depends on the statistic) � 95% of 95% confidence intervals will capture the truth Statistics: Unlocking the Power of Data Lock 5

Confidence Level Suppose you each go out, collect a good random sample of data,

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% Statistics: Unlocking the Power of Data Lock 5

95% of statistics will be within 2 SE of the true parameter value (95%

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

The interval statistic ± 2 SE will include the parameter 95% of the time

The interval statistic ± 2 SE 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%) Statistics: Unlocking the Power of Data Lock 5

95% Confidence Interval If the sampling distribution is relatively symmetric and bell-shaped, a 95%

95% Confidence Interval If the sampling distribution is relatively symmetric and bell-shaped, a 95% confidence interval can be estimated using statistic ± 2 × SE • General form for an interval estimate: statistic ± margin of error • For a 95% confidence interval: margin of error = 2 x SE Statistics: Unlocking the Power of Data Lock 5

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

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

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

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

Interpreting a Confidence Interval � 95% of all samples yield intervals that contain the

Interpreting a Confidence Interval � 95% of all samples yield intervals that contain the true parameter; �We are “ 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 the average time for a trained Golden Shiner to swim to the yellow target is between 46. 2 and 55. 8 seconds. Statistics: Unlocking the Power of Data Lock 5

Interpreting a Confidence Interval confidence level parameter We are 95% confident that the average

Interpreting a Confidence Interval confidence level parameter We are 95% confident that the average time for a trained Golden Shiner to context swim to the yellow target is between 46. 2 and 55. 8 seconds. confidence interval Statistics: Unlocking the Power of Data Lock 5

Interpreting a Confidence Interval �We are [confidence level] confident that the [parameter] [in context]

Interpreting a Confidence Interval �We are [confidence level] confident that the [parameter] [in context] is between [confidence interval]. �“We are 95% confident that the average time for a golden shiner to reach the yellow target is between 46. 2 and 55. 8 seconds. ” Statistics: Unlocking the Power of Data Lock 5

Common Misinterpretations • Misinterpretation 1: “A 95% confidence interval contains 95% of the data

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 the 95% confidence interval” • Misinterpretation 3: “ 95% of all sample means will fall within this 95% confidence interval” • Correct: 95% confident that interval contains the true parameter value! Statistics: Unlocking the Power of Data Lock 5

The Golden Shiner Experiment �Golden shiners have natural preference for yellow, so those trained

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? Statistics: Unlocking the Power of Data Lock 5

Question #2 Does a minority with a stronger opinion or a majority with a

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

Question #2 Does a minority with a stronger opinion or a majority with a

Question #2 Does a minority with a stronger opinion or a majority with a weaker opinion win? � Statistics: Unlocking the Power of Data Lock 5

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

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

Indifferent Fish �Experiment was conducted the same as before, but this time they added

Indifferent Fish �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… a) The strong-opinioned minority b) The weak-opinioned majority Statistics: Unlocking the Power of Data Lock 5

Question #3 What is the effect of indifferent fish on the proportion of times

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

Question #3 What is the effect of indifferent fish on the proportion of times

Question #3 What is the effect of indifferent fish on the proportion of times the majority wins? Statistics: Unlocking the Power of Data Lock 5

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

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

Confidence Intervals � Statistics: Unlocking the Power of Data Lock 5

Confidence Intervals � Statistics: Unlocking the Power of Data Lock 5

Confidence Intervals Confidence Interval Sample Population statistic ± ME Sample . . . Sample

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

Summary • To create a plausible range of values for a parameter: o o

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

To Do �HW 3. 1, 3. 2 (due Monday, 2/13) Statistics: Unlocking the Power

To Do �HW 3. 1, 3. 2 (due Monday, 2/13) Statistics: Unlocking the Power of Data Lock 5