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BNAD 276: Statistical Inference in Management Spring 2016 Green sheets

Before our next exam (March 22 nd ) OpenStax Chapters 1 – 11 Plous (10, 11, 12 & 14) Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability and Risk Chapter 14: The Perception of Randomness Schedule of readings

Homework Assignments On class website: Please print and complete homework worksheet #9 Approaches to probabilities Interpreting probabilities using the normal curve Due: Today Please complete homework worksheet #10 Dan Gilbert Reading - Law of Large Numbers Due: Tuesday, March 1 st

By the end of lecture today 2/25/16 Use this as your study guide Approaches to probability: Empirical, Subjective and Classical Central Limit Theorem

Proposition 1: If sample size (n) is large enough (e.g. 100) The mean of the sampling distribution will approach the mean of the population Central Limit Theorem Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population XXXXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXX XXXXXX XXXX XXXX X X XXXXXXXXXX XXXXXXXXXX X XXXXXXXXXX Proposition 3: The standard deviation of the sampling distribution equals the standard deviation of the population divided by the square root of the sample size. As n increases SEM decreases. As n ↑ x will approach µ As n ↑ curve will approach normal shape As n ↑ curve variability gets smaller

Writing Assignment: Writing a letter to a friend Imagine you have a good friend (pick one). This is a good friend whom you consider to be smart and interested in stuff generally. They are teaching themselves stats (hoping to test out of the class) but need your help on a couple ideas. For this assignment please write your friend/mom/dad/ favorite cousin a letter answering these questions: (Feel free to use diagrams and drawings if you think that can help) Dear Friend, 1. I’m struggling with this whole Central Limit Theorem idea. Could you describe for me the difference between a distribution of raw scores, and a distribution of sample means? 2. I also don’t get the “three propositions of the Central Limit Theorem”. They all seem to address sample size, but I don’t get how sample size could affect these three things. If you could help explain it, that would be really helpful. We’ll start this in class and finish as part of homework

Imagine you have a good friend (pick one). This is a good friend whom you consider to be smart and interested in stuff generally. They are teaching themselves stats (hoping to test out of the class) but need your help on a couple ideas. For this assignment please write your friend/mom/dad/ favorite cousin a letter answering these questions: (Feel free to use diagrams and drawings if you think that can help) Dear Friend, 1. I’m struggling with this whole Central Limit Theorem idea. Could you describe for me the difference between a distribution of raw scores, and a distribution of sample means? 2. I also don’t get the “three propositions of the Central Limit Theorem”. They all seem to address sample size, but I don’t get how sample size could affect these three things. If you could help explain it, that would be really helpful.

Review of Homework Worksheet just in case of questions

Homework review Based on apriori probability – all options equally likely – not based on previous experience or data Based on expert opinion - don’t have previous data for these two companies merging together 2 5 =.40 Based on frequency data (Percent of rockets that successfully launched)

Homework review Based on apriori probability – all options equally likely – not based on previous experience or data Based on frequency data (Percent of times that pages that are “fake”) =.30 Based on frequency data (Percent of times at bat that successfully resulted in hits)

Homework review 5 50 =.10 Based on frequency data (Percent of students who successfully chose to be Economics majors)

= = z of 1.5 = area of =.8276 z of 1.25 = area of = = = area of = = z of.5 = area of =.2029 z of 1.25 = area of

Homework review = 0.45 z of 0.45 = area of = = 0.45 z of 0.45 = area of = = 1.22 z of 1.22 = area of = -.32 z of = area of = = 1.22 z of 1.22 = area of

Homework review = 1.43 z of 1.43 = area of = = 1.43 z of 1.43 = area of.4236 z of = area of –.3051 =.1185 z of -.86 = area of = = =

Now we have all the pieces we need to create confidence intervals

Confidence intervals: defining most common scores Combining (and reviewing) skills to build confidence intervals What confidence intervals are most useful for Finding scores that border the middle 95% of curve Using what we know about Central Limit Theorem to use the standard error of the mean rather than the standard deviation Confidence Interval: We are estimating a value but providing two scores between which we believe the true value lies. We can be 95% confident that our mean falls between these two scores.

Confidence intervals: defining most common scores Step 1 review what confidence intervals are most useful for Confidence Interval: We are estimating a value by providing two scores between which we believe the true value lies. We can be 95% confident that our mean falls between these two scores.

95% Confidence Interval: We can be 95% confident that our population mean falls between these two scores 99% Confidence Interval: We can be 99% confident that our population mean falls between these two scores Confidence Intervals: What are they used for? We are estimating a value by providing two scores between which we believe the true value lies. We can be 95% confident that our mean falls between these two scores.

We are using this to estimate a value such as a mean, with a known degree of certainty with a range of values The interval refers to possible values of the population mean. We can be reasonably confident that the population mean falls in this range (90%, 95%, or 99% confident) In the long run, series of intervals, like the one we figured out will describe the population mean about 95% of the time. Greater confidence implies loss of precision. (95% confidence is most often used) Confidence Intervals: What are they used for? Can actually generate CI for any confidence level you want – these are just the most common Subjective vs Empirical

Confidence intervals Step 2 review how we find the scores that border the middle 95% of curve

Raw Scores Area & Probability Z Scores Formula z table Have raw score Find z Have z Find raw score Have area Find z Have z Find area Normal distribution Raw scores z-scores probabilities Building towards confidenc e intervals Par t 1

. Try this one: Please find the (2) raw scores that border exactly the middle 95% of the curve Mean of 30 and standard deviation of 2 Go to table.4750 nearest z = 1.96 mean + z σ = 30 + (1.96)(2) = Go to table.4750 nearest z = mean + z σ = 30 + (-1.96)(2) = Building towards confidenc e intervals Par t 1

. Try this one: Please find the (2) raw scores that border exactly the middle 99% of the curve Mean of 30 and standard deviation of 2 Go to table.4950 nearest z = 2.58 mean + z σ = 30 + (2.58)(2) = Go to table.4950 nearest z = mean + z σ = 30 + (-2.58)(2) = Building towards confidenc e intervals Par t 1

. Please find the raw scores that border the middle 99% of the curve Please find the raw scores that border the middle 95% of the curve Which is wider? Building towards confidenc e intervals Par t 1

. Please find the raw scores that border the middle 99% of the curve Please find the raw scores that border the middle 95% of the curve 95% Confidence Interval: We can be 95% confident that the estimated score really does fall between these two scores 99% Confidence Interval: We can be 99% confident that the estimated score really does fall between these two scores Building towards confidenc e intervals Par t 1

1) Go to z table - find z score for for area.4750 z = 1.96 Mean = 50 Standard deviation = 10 Find the scores that border the middle 95% ? ? ? 95% 2) x = mean + (z)(standard deviation) x = 50 + (-1.96)(10) x = ) x = mean + (z)(standard deviation) x = 50 + (1.96)(10) x = Please note: We will be using this same logic for “confidence intervals” x = mean ± (z)(standard deviation) Scores capture the middle 95% of the curve ?

Confidence intervals Step 3 review how central limit theorem affects how we use samples and how sample size can affect our estimation

Confidence Intervals: A range of values that, with a known degree of certainty, includes an estimated value (like a mean) How can we make our confidence interval smaller? Increase sample size (This will decrease variability) Decrease level of confidence Decrease variability through more careful assessment and measurement practices (minimize noise). 95%

Mean = 50 Standard deviation = 10 n = 100 s.e.m. = 1 Construct a 95% confidence interval ? ? ? 95% x = mean ± (z)(s.e.m.) x = 50 + (1.96)(1) x = % Confidence Interval is captured by the scores – For “confidence intervals” same logic – same z-score But - we’ll replace standard deviation with the standard error of the mean standard error of the mean σ n = = x = 50 + (-1.96)(1) x = 48.04

mean = 121 standard deviation= 15 n = standard error of the mean σ n = raw score = mean + (z score)(standard error) x = x ± ( z )( σ x ) raw score = mean ± (z score)(sem) =3 = X = 121 ± (1.96)(3) = 121 ± 5.88 (115.12, ) confidence interval Please notice: We know the standard deviation and can calculate the standard error of the mean from it Find a 95% Confidence Interval for this distribution

We used this one when finding raw scores associated with an area under the curve. We had all population info. Not really a “confidence interval” because we know the mean of the population, so there is nothing to estimate or be “confident about”. Construct a 95 percent confidence interval around the mean Some Mean Some Variability Hint always draw a picture! ?? 95% We used this one when finding raw scores associated with an area under the curve. We used this to provide an interval within which we believe the mean falls. We have some level of confidence about our guess. We know the population standard deviation. We know this Similar, but uses standard error the mean based on population s.d. raw score = mean ± (z score)(s.d.) raw score = mean ± (z score)(s.e.m.)

Confidence intervals Tell me the scores that border exactly the middle 95% of the curve - use z score of 1.96 Construct a 95 percent confidence interval around the mean z scores for different levels of confidence Level of Alpha 1.96 = = =.10 90% How do we know which z score to use?

Confidence Interval of 99% Has and alpha of 1% α =.01 Confidence Interval of 90% Has and alpha of 10% α =. 10 Confidence Interval of 95% Has and alpha of 5% α =.05 99%95%90% Area outside confidence interval is alpha Area in the tails is called alpha Area associated with most extreme scores is called alpha Critical z Critical z 2.58 Critical z Critical z 1.96 Critical z Critical z 1.64