Political Science 30: Political Inquiry Section 5
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1 Political Science 30: Political Inquiry Section 5 Taylor Carlson tncarlson@ucsd.edu Link to Stats Motivation of the Week They ve done studies, you know. 60% of the time, it works every time. Brian, Anchorman Carlson POLI 30-Section 5 February 16, / 17
2 Learning Outcomes By the end of section today, you should: Be able to calculate and interpret: Margin of Error 95% Confidence Interval Proportion (nominal data) Mean (interval/continuous data) Explain the intuition behind a significance test Feel prepared to tackle Homework 2 Carlson POLI 30-Section 5 February 16, / 17
3 Warm Up Name Reflect on the midterm. How do you feel it went for you? What strategies were effective? What will you do differently when studying for the final? Was there anything covered on the midterm that you still find confusing? Carlson POLI 30-Section 5 February 16, / 17
4 Announcements If you have questions on HW1, please come see me in office hours, especially if with Question 3 since you build on this for HW 2-4 Carlson POLI 30-Section 5 February 16, / 17
5 Announcements If you have questions on HW1, please come see me in office hours, especially if with Question 3 since you build on this for HW 2-4 See sample answer key for HW1 on TritonEd Carlson POLI 30-Section 5 February 16, / 17
6 Announcements If you have questions on HW1, please come see me in office hours, especially if with Question 3 since you build on this for HW 2-4 See sample answer key for HW1 on TritonEd HW 2 is due February 26th at the beginning of lecture Hard copy and Turn it In Please staple! You ve already learned how to do all parts on this homework! Carlson POLI 30-Section 5 February 16, / 17
7 Announcements If you have questions on HW1, please come see me in office hours, especially if with Question 3 since you build on this for HW 2-4 See sample answer key for HW1 on TritonEd HW 2 is due February 26th at the beginning of lecture Hard copy and Turn it In Please staple! You ve already learned how to do all parts on this homework! The course moves pretty quickly from here: Carlson POLI 30-Section 5 February 16, / 17
8 Announcements If you have questions on HW1, please come see me in office hours, especially if with Question 3 since you build on this for HW 2-4 See sample answer key for HW1 on TritonEd HW 2 is due February 26th at the beginning of lecture Hard copy and Turn it In Please staple! You ve already learned how to do all parts on this homework! The course moves pretty quickly from here: HW2 due Feb 26 HW3 due March 7 HW4 due March 14 Final exam March 23 Carlson POLI 30-Section 5 February 16, / 17
9 My Advice: Stay on top of the readings, including the textbook Do the practice problems in the book each week (odds have answers in the back of the book) Do the practice problems I post online Come to lecture and section Come see us in office hours if you don t understand something Start the homework assignments early! Carlson POLI 30-Section 5 February 16, / 17
10 Quick Review: Normal Curves What do all normal curves have in common? Carlson POLI 30-Section 5 February 16, / 17
11 Quick Review: Normal Curves What do all normal curves have in common? Symmetric Mean = Median Bell-shaped, with most of their density at the center Carlson POLI 30-Section 5 February 16, / 17
12 Quick Review: Normal Curves Carlson POLI 30-Section 5 February 16, / 17
13 Quick Review: Central Limit Theorem (CLT) If we draw lots and lots of samples and calculate the mean of each sample, these sample means will be normally distributed around the true population mean. Carlson POLI 30-Section 5 February 16, / 17
14 Quick Review: Central Limit Theorem (CLT) If we draw lots and lots of samples and calculate the mean of each sample, these sample means will be normally distributed around the true population mean. This means, that we can use the sample mean and the properties of the normal distribution to determine the range in which the true population mean resides. Carlson POLI 30-Section 5 February 16, / 17
15 Quick Review: Central Limit Theorem (CLT) If we draw lots and lots of samples and calculate the mean of each sample, these sample means will be normally distributed around the true population mean. This means, that we can use the sample mean and the properties of the normal distribution to determine the range in which the true population mean resides. That range is a... Carlson POLI 30-Section 5 February 16, / 17
16 Confidence Interval Carlson POLI 30-Section 5 February 16, / 17
17 Confidence Interval A confidence interval tells us the range of values in which the true population parameter resides at a given level of confidence (e.g. 95%, 99.7%). Carlson POLI 30-Section 5 February 16, / 17
18 Confidence Interval Carlson POLI 30-Section 5 February 16, / 17
19 Confidence Interval What kind of data do we have? Carlson POLI 30-Section 5 February 16, / 17
20 Confidence Interval What kind of data do we have? Nominal Carlson POLI 30-Section 5 February 16, / 17
21 Confidence Interval What kind of data do we have? Nominal Use proportions ˆp ± 2 (ˆp)(1 ˆp) N Carlson POLI 30-Section 5 February 16, / 17
22 Confidence Interval What kind of data do we have? Nominal Use proportions ˆp ± 2 Interval / Continuous (ˆp)(1 ˆp) N Carlson POLI 30-Section 5 February 16, / 17
23 Confidence Interval What kind of data do we have? Nominal Use proportions ˆp ± 2 Interval / Continuous Use means X ± 2 ˆσ N (ˆp)(1 ˆp) N Carlson POLI 30-Section 5 February 16, / 17
24 Confidence Interval of a Proportion (Nominal Data) Awful Announcing recently conducted a poll on Twitter about whether individuals prefer the Winter or Summer Olympics. 2,353 individuals responded to the poll, and the results show that 36 percent prefer the Winter Olympics and 64 percent prefer the Summer Olympics. Assume that the respondents were a representative sample of Twitter Users. Calculate the Margin of Error and the 95% Confidence Interval for the proportion of Twitter Users who prefer the Winter Olympics. Carlson POLI 30-Section 5 February 16, / 17
25 Confidence Interval of a Proportion (Nominal Data) Calculate the Margin of Error: Carlson POLI 30-Section 5 February 16, / 17
26 Confidence Interval of a Proportion (Nominal Data) Calculate the Margin of Error: 2 (ˆp)(1 ˆp) N Carlson POLI 30-Section 5 February 16, / 17
27 Confidence Interval of a Proportion (Nominal Data) Calculate the Margin of Error: 2 2 (ˆp)(1 ˆp) N (.36)(1.36) 2353 Carlson POLI 30-Section 5 February 16, / 17
28 Confidence Interval of a Proportion (Nominal Data) Calculate the Margin of Error: (ˆp)(1 ˆp) N (.36)(1.36) Carlson POLI 30-Section 5 February 16, / 17
29 Confidence Interval of a Proportion (Nominal Data) Calculate the Margin of Error: (ˆp)(1 ˆp) N (.36)(1.36) Carlson POLI 30-Section 5 February 16, / 17
30 Confidence Interval of a Proportion (Nominal Data) Calculate the 95% Confidence Interval: Carlson POLI 30-Section 5 February 16, / 17
31 Confidence Interval of a Proportion (Nominal Data) ˆp ± 2 Calculate the 95% Confidence Interval: (ˆp)(1 ˆp) N Carlson POLI 30-Section 5 February 16, / 17
32 Confidence Interval of a Proportion (Nominal Data) ˆp ± 2.36 ± 2 (0.01) Calculate the 95% Confidence Interval: (ˆp)(1 ˆp) N Carlson POLI 30-Section 5 February 16, / 17
33 Confidence Interval of a Proportion (Nominal Data) ˆp ± 2.36 ± 2 (0.01).36 ± 0.02 Calculate the 95% Confidence Interval: (ˆp)(1 ˆp) N Carlson POLI 30-Section 5 February 16, / 17
34 Confidence Interval of a Proportion (Nominal Data) ˆp ± 2.36 ± 2 (0.01).36 ± ,.38 Calculate the 95% Confidence Interval: (ˆp)(1 ˆp) N Carlson POLI 30-Section 5 February 16, / 17
35 Confidence Interval of a Proportion (Nominal Data) ˆp ± 2.36 ± 2 (0.01).36 ± ,.38 Calculate the 95% Confidence Interval: (ˆp)(1 ˆp) N Interpretation: We can conclude with 95% confidence that the true population proportion of Twitter users who prefer the Winter Olympics is between.34 and.38. Carlson POLI 30-Section 5 February 16, / 17
36 Confidence Interval of a Mean (Interval Data) Chapter 6, Problem 8 We collect data from the FEC (Federal Election Committee) cataloguing all individual contributions to a given presidential primary campaign. The mean is $575 and the standard deviation is 250. If the sample size was 625 and we wanted to use a 95% confidence interval, what would we be able to confidently claim is the true mean contribution level for the population? Confidence Interval Formula: X ± 2 ˆσ N Carlson POLI 30-Section 5 February 16, / 17
37 Confidence Interval of a Mean (Interval Data) 575 ± ± ± ± , 595 Interpretation: We can conclude with 95% confidence that the true population mean is between 555 and 595. Carlson POLI 30-Section 5 February 16, / 17
38 How might we turn these into significance tests? Let s go back to the Olympics example. Expectation: The minority of Twitter users prefer the Winter Olympics. Carlson POLI 30-Section 5 February 16, / 17
39 How might we turn these into significance tests? Let s go back to the Olympics example. Expectation: The minority of Twitter users prefer the Winter Olympics. Null Hypothesis (H 0 ): the minority of Twitter users do not prefer the Winter Olympics Carlson POLI 30-Section 5 February 16, / 17
40 How might we turn these into significance tests? Let s go back to the Olympics example. Expectation: The minority of Twitter users prefer the Winter Olympics. Null Hypothesis (H 0 ): the minority of Twitter users do not prefer the Winter Olympics ˆp PrefersWinter 0.5 Carlson POLI 30-Section 5 February 16, / 17
41 How might we turn these into significance tests? Let s go back to the Olympics example. Expectation: The minority of Twitter users prefer the Winter Olympics. Null Hypothesis (H 0 ): the minority of Twitter users do not prefer the Winter Olympics ˆp PrefersWinter 0.5 Alternative Hypothesis (H 1 ): the minority of Twitter users prefer the Winter Olympics Carlson POLI 30-Section 5 February 16, / 17
42 How might we turn these into significance tests? Let s go back to the Olympics example. Expectation: The minority of Twitter users prefer the Winter Olympics. Null Hypothesis (H 0 ): the minority of Twitter users do not prefer the Winter Olympics ˆp PrefersWinter 0.5 Alternative Hypothesis (H 1 ): the minority of Twitter users prefer the Winter Olympics ˆp PrefersWinter < 0.5 Carlson POLI 30-Section 5 February 16, / 17
43 How might we turn these into significance tests? Check the data what does our observed proportion tell us? Carlson POLI 30-Section 5 February 16, / 17
44 How might we turn these into significance tests? Check the data what does our observed proportion tell us? 36% of Twitter users prefer the Winter Olympics, which is a minority. Carlson POLI 30-Section 5 February 16, / 17
45 How might we turn these into significance tests? Check the data what does our observed proportion tell us? 36% of Twitter users prefer the Winter Olympics, which is a minority. But, how sure are we that we re right? Carlson POLI 30-Section 5 February 16, / 17
46 How might we turn these into significance tests? Check the data what does our observed proportion tell us? 36% of Twitter users prefer the Winter Olympics, which is a minority. But, how sure are we that we re right? Recall the confidence interval we calculated earlier: (.34,.38) Carlson POLI 30-Section 5 February 16, / 17
47 How might we turn these into significance tests? Check the data what does our observed proportion tell us? 36% of Twitter users prefer the Winter Olympics, which is a minority. But, how sure are we that we re right? Recall the confidence interval we calculated earlier: (.34,.38) Recall that our null hypothesis is ˆp PrefersWinter 0.5 Carlson POLI 30-Section 5 February 16, / 17
48 How might we turn these into significance tests? Check the data what does our observed proportion tell us? 36% of Twitter users prefer the Winter Olympics, which is a minority. But, how sure are we that we re right? Recall the confidence interval we calculated earlier: (.34,.38) Recall that our null hypothesis is ˆp PrefersWinter 0.5 Is 0.50 inside our confidence interval? Carlson POLI 30-Section 5 February 16, / 17
49 How might we turn these into significance tests? Check the data what does our observed proportion tell us? 36% of Twitter users prefer the Winter Olympics, which is a minority. But, how sure are we that we re right? Recall the confidence interval we calculated earlier: (.34,.38) Recall that our null hypothesis is ˆp PrefersWinter 0.5 Is 0.50 inside our confidence interval? If YES FAIL TO REJECT the null hypothesis If NO REJECT the null hypothesis Carlson POLI 30-Section 5 February 16, / 17
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