Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Spring 2017 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays. Welcome http://www.youtube.com/watch?v=oSQJP40PcGI http://www.youtube.com/watch?v=oSQJP40PcGI

A note on doodling

By the end of lecture today 2/24/17 Confidence Intervals Central Limit Theorem

Due: Wednesday, March 1st Homework Assignment 15 & 16 Please complete the homework modules on the D2L website HW15-Confidence Intervals Please complete this homework worksheet 16 Confidence Intervals Due: Wednesday, March 1st

Before next exam (March 3rd) Schedule of readings Before next exam (March 3rd) Please read chapters 1 - 8 in OpenStax textbook Please read Chapters 10, 11, 12 and 14 in Plous Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability and Risk Chapter 14: The Perception of Randomness

Lab sessions Everyone will want to be enrolled in one of the lab sessions Labs Exam 2 Prep

Used in lots of different contexts. Sometimes with different names Confidence Intervals We need three foundational building blocks to create confidence intervals Used in lots of different contexts. Sometimes with different names What is it for? How to Find it Why SEM?

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

What are they for? Confidence Intervals: Combining these three skills to build confidence intervals What are they for? Confidence Intervals: 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. Central Limit Theorem Central Limit Theorem highlights importance of standard error of the mean (which is built on standard deviation) Using standardized scores (z scores) to find raw score values that border the middle part of the curve. Building towards confidence intervals

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

Writing Assignment: Part 1 . Writing Assignment: Part 1

? ? .9500 .475 .475 . 28 32 30 Writing Assignment: Part 1 Question 1: Please find the (2) raw scores that exactly border the middle 95% of the curve Mean of 30 and standard deviation of 2 . .9500 .475 .475 Building towards confidence intervals ? 28 30 32 ? Part 1

? ? .9900 .495 .495 . 28 32 30 Writing Assignment: Part 1 Question 2: Please find the (2) raw scores that exactly border the middle 99% of the curve Mean of 30 and standard deviation of 2 . .9900 .495 .495 Building towards confidence intervals ? 28 30 32 ? Part 1

? ? .9500 .475 .475 . 25 35 30 Writing Assignment: Part 1 Question 3: Please find the (2) raw scores that exactly border the middle 95% of the curve Mean of 30 and standard deviation of 5 . .9500 .475 .475 Building towards confidence intervals ? 25 30 35 ? Part 1

Compare your results from questions 1 and 2. Which interval is wider and why? Question 1: Please find the raw scores that border the middle 95% of the curve Question 2: Building towards confidence intervals Please find the raw scores that border the middle 99% of the curve Part 1

Compare your results from questions 1 and 3. Which interval is wider and why? Question 1: Please find the raw scores that border the middle 95% of the curve Mean of 30 standard deviation of 2 . Question 3: Please find the raw scores that border the middle 95% of the curve Mean of 30 standard deviation of 5 Building towards confidence intervals Part 1

Question 6: What are the two ways to make the confidence interval smaller (narrower)? . 95% 95%

Writing Assignment: Part 1 . Writing Assignment: Part 1 Solutions

. Question 1: Please find the (2) raw scores that exactly border 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) = 33.92 Go to table .4750 nearest z = -1.96 mean + z σ = 30 + (-1.96)(2) = 26.08 .9500 .475 .475 Building towards confidence intervals 26.08 ? 33.92 28 32 ? 30 Part 1

. Question 2: Please find the (2) raw scores that exactly border 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) = 35.16 Go to table .4950 nearest z = -2.58 mean + z σ = 30 + (-2.58)(2) = 24.84 .9900 .495 .495 Building towards confidence intervals 24.84 ? 35.16 28 32 ? 30 Part 1

. Question 3: Please find the (2) raw scores that exactly border the middle 95% of the curve Mean of 30 and standard deviation of 5 Go to table .4750 nearest z = 1.96 mean + z σ = 30 + (1.96)(5) = 39.8 Go to table .4750 nearest z = -1.96 mean + z σ = 30 + (-1.96)(5) = 20.2 .9500 .475 .475 Building towards confidence intervals 20.2 ? 39.8 28 32 ? 30 Part 1

Compare your results from questions 1 and 2. Which interval is wider and why? 26.08 33.92 Please find the raw scores that border the middle 95% of the curve 24.84 35.16 Building towards confidence intervals Please find the raw scores that border the middle 99% of the curve Part 1

Compare your results from questions 1 and 3. Which interval is wider and why? 26.08 33.92 Please find the raw scores that border the middle 95% of the curve Mean of 30 standard deviation of 2 20.2 39.8 Please find the raw scores that border the middle 95% of the curve Mean of 30 standard deviation of 5 Building towards confidence intervals Part 1

Question 6: What are the two ways to make the confidence interval smaller (narrower)? Decrease level of confidence Decrease variability (make standard deviation smaller) 1. Increase sample size (This will decrease variability) 2. Very careful assessment and measurement practices (improve reliability will minimize noise) . 95% 95%

Remember Confidence Intervals . Remember Confidence Intervals 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 confidence intervals Part 1

Calculating Two Scores that Border Middle 95% of curve Step 1: Find mean (expected value) x is mean of sample Step 2: Find standard deviation and standard error of the mean “σ” if population Step 3: Decide on area of interest - Middle 95% α = .05  z = 1.96 - Middle 99% α = .01  z = 2.58 Step 4: Calculations for confidence intervals Step 5: Conclusion - tie findings with statement about what you are estimating

95% ? ? Find the scores that border the middle 95% Mean = 50 Standard deviation = 10 Find the scores that border the middle 95% ? ? 95% x = mean ± (z)(standard deviation) 30.4 69.6 .9500 .4750 .4750 Please note: We will be using this same logic for “confidence intervals” ? ? 1) Go to z table - find z score for for area .4750 z = 1.96 2) x = mean + (z)(standard deviation) x = 50 + (-1.96)(10) x = 30.4 30.4 3) x = mean + (z)(standard deviation) x = 50 + (1.96)(10) x = 69.6 69.6 Scores 30.4 - 69.6 capture the middle 95% of the curve

Calculating Confidence Intervals Step 1: Find mean (expected value) x is mean of sample Step 2: Find standard deviation and standard error of the mean “σ” if population Step 3: Decide on level of confidence - 95% Confident α = .05  z = 1.96 - 99% Confident α = .01  z = 2.58 Step 4: Calculations for confidence intervals Step 5: Conclusion - tie findings with statement about what you are estimating

σ n 95% ? ? 10 = = √ 100 √ Construct a 95% confidence interval Mean = 50 Standard deviation = 10 n = 100 s.e.m. = 1 ? ? 95% 48.04 51.96 ? .9500 .4750 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 = 10 = x = mean ± (z)(s.e.m.) √ 100 x = 50 + (1.96)(1) x = 51.96 x = 50 + (-1.96)(1) x = 48.04 95% Confidence Interval is captured by the scores 48.04 – 51.96

? ? Construct a 95 percent confidence interval around the mean 95% ? We know this raw score = mean ± (z score)(s.d.) Some Mean Some Variability 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”. Hint always draw a picture! 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. raw score = mean ± (z score)(s.e.m.) Similar, but uses standard error the mean based on population s.d.

Confidence interval uses SEM Homework Worksheet: Confidence interval uses SEM

Thank you! See you next time!!