Inference for Categorical Variable: Confidence Intervals - Lecture Slides | STAT 0200, Study notes of Statistics

Download Inference for Categorical Variable: Confidence Intervals - Lecture Slides | STAT 0200 and more Study notes Statistics in PDF only on Docsity! (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 1 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture Lecture 21 Inference for Categorical Variable: Confidence Intervals 3 Forms of Inference Probability vs. Confidence Constructing Confidence Interval Sample Size; Level of Confidence (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.2 Looking Back: Review  4 Stages of Statistics  Data Production (discussed in Lectures 1-4)  Displaying and Summarizing (Lectures 5-12)  Probability (discussed in Lectures 13-20)  Statistical Inference  1 categorical  1 quantitative  categorical and quantitative  2 categorical  2 quantitative (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.3 Four Processes of Statistics Sample 1. PRODUCE DATA 2. SUMMARIZE 3. PROBABILITY 4. INFERENCE C Q CQ CC QQ Population Assume we know what’s true for the population; how should random samples behave? Assume we only know what’s true for the sample; what can we infer about the larger population? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.4 Summarizing Categorical Sample Data (Review) What proportion of sampled students ate breakfast the day of the survey? Looking Back: In Part 2, we summarized sample data for single variables or relationships. Looking Ahead: In Part 4, our goal is to go beyond sample data and draw conclusions about the larger population from which the sample was obtained. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 2 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.5 Three Types of Inference Problem In a sample of 446 students, 0.55 ate breakfast. 1. What is our best guess for the population proportion of students who eat breakfast? Point Estimate 2. What interval should contain the population proportion of students who eat breakfast? Confidence Interval 3. Is the population proportion of students who eat breakfast more than half (50%)? Hypothesis Test (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.6 Behavior of Sample Proportion (Review) For random sample of size n from population with p in category of interest, sample proportion has  mean p  is unbiased estimator of p (sample must be random) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.7 Example: Checking if Estimator is Unbiased  Background: Survey produced sample proportion of intro stat students (various ages and times of day) at a university who’d eaten breakfast.  Questions:  Is the sample representative of all college students? All students at that university?  Were the values of the variable (breakfast or not) recorded without bias? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.9 Example: Checking if Estimator is Unbiased  Background: Survey produced sample proportion of intro stat students (of various years, in classes meeting at various times of day) at a university who’d eaten breakfast.  Responses:  Differences among college cafeterias, etc.  Question not sensitive (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 5 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.22 Three Types of Inference Problem In a sample of 446 students, 0.55 ate breakfast. 1. What is our best guess for the population proportion of students who eat breakfast? Point Estimate 2. What interval should contain the population proportion of students who eat breakfast? Confidence Interval 3. Do more than half (50%) of the population of students eat breakfast? Hypothesis Test (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.23 Beyond a Point Estimate Sample proportion from unbiased sample is best estimate for population proportion. Looking Ahead: For point estimate we don’t need sample size or info about spread. These are required for confidence intervals and hypothesis tests, to quantify how good our point estimate is. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.25 Probability vs. Confidence  Probability: given population proportion, how does sample proportion behave?  Confidence: given sample proportion, what is a range of plausible values for population proportion? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.26 Example: Probability Statement  Background: If students pick numbers from 1 to 20 at random, p=0.05 should pick #7. For n=400, has  mean 0.05  s.d  shape approx. normal.  Question: What does the “95” part of the 68-95-99.7 Rule tell us about ? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 6 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.28 Example: Probability Statement  Background: If students pick numbers from 1 to 20 at random, p=0.05 should pick #7. For n=400, has  mean 0.05  s.d = 0.01  shape approx. normal.  Response: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.29 Example: Probability Statement Looking Ahead: This statement about sample proportion is correct but not very useful for practical purposes. In most real-life problems, we want to draw conclusions about an unknown population proportion. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.30 Example: How Far is One from the Other?  Background: Suppose a friend is passing through town and calls to say, “I’m within half a mile of your house.”  Question: What can be said about where your house is in relation to the friend? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.32 Example: How Far is One from the Other?  Background: Suppose a friend is passing through town and calls to say, “I’m within half a mile of your house.”  Response: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 7 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.33 Definitions Margin of Error: Distance around a sample statistic, within which we have reason to believe the corresponding parameter falls. A common margin of error is 2 s.d.s. Confidence Interval for parameter: Interval within which we have reason to believe the parameter falls = range of plausible values A common confidence interval is sample statistic plus or minus 2 s.d.s. A Closer Look: A parameter is not a R.V.does not obey laws of probabilityuse word“confidence”. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.34 Example: Confidence Interval for p  Background:30/400=0.075 students picked #7 “at random” from 1 to 20. Let’s assume sample proportion for n=400 has s.d. 0.01.  Question: What can we claim about population proportion p picking #7? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.36 Example: Confidence Interval for p  Background:30/40=0.075 students picked #7 “at random” from 1 to 20. Let’s assume sample proportion for n=400 has s.d. 0.01.  Response: Looking Back: My probability statement claimed sample proportion should fall within 2 s.d.s of population proportion. Now, the inference statement claims population proportion should be within 2 s.d.s of sample proportion. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.37 Example: Confidence Interval for p By “pretty sure”, we mean “95% confident”, because 95% is the probability of sample proportion within 2 s.d.s of p (for large enough n). (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 10 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.46 Conditions for Normality in Confidence Interval Multiplier 2 from normal dist. approximately correct if np and n(1-p) both at least 10. But p is unknown so substitute : Require Sample count in (X) and out (n-X) of category of interest should both be at least 10. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.47 Example: Checking Sample Size  Background: 30/400=0.075 students picked #7 “at random” from 1 to 20.  Question: Do the data satisfy requirement for approximate normality of sample proportion? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.49 Example: Checking Sample Size  Background: 30/400=0.075 students picked #7 “at random” from 1 to 20.  Response: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.50 Example: Checking Population Size  Background: To draw conclusions about criminal histories of a city’s 750 bus drivers, a random sample of 100 drivers was used.  Question: Is there approximate independence in spite of sampling without replacement, so formula for standard error is accurate? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 11 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.52 Example: Checking Population Size  Background: To draw conclusions about criminal histories of a city’s 750 bus drivers, a random sample of 100 drivers was used.  Response: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.53 Example: Revisiting Original Question  Background: In sample of 446 college students, 246 (proportion 0.55) ate breakfast.  Question: Assuming sample is representative, what interval should contain proportion of all students at that university who eat breakfast? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.56 Example: Revisiting Original Question  Background: In sample of 446 college students, 246 (proportion 0.55) ate breakfast.  Response: Approx. 95% confidence interval for p is =___________________________________ Looking Back: Earlier we wondered if a majority of students eat breakfast. The interval suggests this is the case, since it is entirely above 0.50. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.57 Example: Role of Sample Size  Background: 95% confidence intervals are constructed based on sample proportion 0.54 from various sample sizes.  Question: What happens as n increases? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 12 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.59 Example: Role of Sample Size  Background: 95% confidence intervals are constructed based on sample proportion 0.54 from various sample sizes.  Response: larger n__________________ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.61 Example: Role of Sample Size  Response: larger n_________________ A Closer Look: 1000 is a common sampling poll size, with margin of error ± 3% (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.62 Other Levels of Confidence Confidence level 95% uses multiplier 2. Other levels use other multipliers, based on normal curve: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L21.63 Other Levels of Confidence Confidence level 95% uses multiplier 2. Other levels use other multipliers, based on normal curve. More precise multiplier for 95% is 1.96 instead of 2.