Understanding Pop. Parameters & Sample Stats: Confidence Intervals & Sampling, Study Guides, Projects, Research of Statistics

Download Understanding Pop. Parameters & Sample Stats: Confidence Intervals & Sampling and more Study Guides, Projects, Research Statistics in PDF only on Docsity! 1 Questions about the Assignment If your answer is wrong, but you show your work you can get more partial credit. Population parameter versus sample statistic Uncertainty in estimates Sampling distribution Confidence interval Understanding Inference: Confidence Intervals I Population Sample Sampling Statistical Inference The Big Picture A sample statistic is a number computed from sample data. (e.g., sample mean: mean income of the people in the sample) A population parameter is a number that describes some aspect of a population. (e.g., population mean: mean income of the entire population) We usually have a sample statistic and want to make inferences about the population parameter. Statistic vs. Parameter Population Sample Sampling Statistical Inference The Big Picture Statistic Parameter Statistic vs. Parameter Sample Statistics Population Parameters Mean μ (mu) Proportion ̂ p Std. Deviation s (sigma) Correlation r ρ (rho) Slope b β (beta) 2 Gallup surveyed 1,500 Americans between June 9th-11th 2012 and 49% of these people approved of the job Barack Obama is doing as president. What is the population? What is the sample size? Is this categorical or quantitative variable? For categorical variables, what sample statistic are we interested in? Sample statistic: (sample proportion) Based on this sample statistic, what do you think is the true proportion of Americans who approve of the job Barack Obama is doing as president? Population parameter: (population proportion) Obama’s Approval Rating http://www.gallup.com/poll/113980/Gallup-Daily-Obama-Job-Approval.aspx ~330million (All Americans) 1,500 Categorical ̂ = .49 Sample proportion p = ? The sample statistic gives a point estimate (a single number) for the population parameter. Usually, it is more useful to provide an interval estimate which gives a range of plausible values for the population parameter: interval estimate = point estimate margin of error How do we determine the margin of error??? Point and Interval Estimates Point Estimate: = .49 Interval Estimate: 0.49 0.03 = (0.46, 0.52) Between 46% and 52% of Americans currently approve of the job Obama is doing as president. Obama’s Approval Rating point estimate margin of error The population parameter is a fixed value. Sample statistics vary from sample to sample. They will not match the population parameter exactly. For a given sample statistic, what are plausible values for the population parameter? How much uncertainty surrounds the sample statistic? It depends on how much the sample statistic varies from sample to sample! Important Points What proportion of Reese’s pieces are orange? Reese’s Pieces 5 A 95% confidence interval can be created by: sample statistic 2 standard deviations point estimate margin of error The point estimate is calculated from our sample. The margin of error is calculated from the sampling distribution. Confidence Intervals The standard deviation of the sampling distribution (i.e., the distribution of sample statistics) is called the standard error (SE). This is done to clearly distinguish it from the standard deviation of the sample distribution. Standard Error: The Standard Deviation of the Sampling Distribution To create a plausible range of values for a parameter: 1. Take many random samples from the population, and compute the sample statistic for each sample. 2. Compute the standard error as the standard deviation of all these statistics. 3. Use: sample statistic ± 2 × standard error One small problem… Often we only have one sample! How can we calculate the variation in sample statistics, if we only have one sample? Summary Part I: Graded Problems 3.12, 3.16, 3.24, and 3.54 Part II: (Type up this assignment in a Word document) Goto http://sda.berkeley.edu/cgi-bin/hsda?harcsda+gss10 Find 3 quantitative variables and for each variable find another quantitative variable that you think is associated with it. Conduct a correlation test to see how correlated they are. For each pair of variables provide the following information: Variable names Question related to the variable Explain in your own words what this variable is measuring The unit used to measure the variable (e.g., years, dollars, inches, etc.) Min, Max, Mean, Median, Standard Deviation (Std Dev) The correlation score An interpretation of the correlation score Assignment Calculating Correlations from the GSS Under the “Analysis” tab, click on the “Correlation matrix” tab. Enter the names of two quantitative variables here. Click on this button and the correlation statistics will open up in a new window. Calculating Correlations from the GSS This is what will pop up in the new window. This is the correlation (r) score for the two variables 6 A recent survey of 1,502 Americans in found that 86% consider the economy a “top priority” for the president and congress this year. The standard error for this statistic is 0.01. What is the 95% confidence interval for the true proportion of all Americans that consider the economy a “top priority” for the president and congress this year? A. (0.85, 0.87) B. (0.84, 0.88) C. (0.82, 0.90) Economy 0.86  2 × 0.01 The standard error of a sample statistic is the same thing as the standard deviation of the sampling distribution (i.e., distribution of sample statistics). In order to calculate the standard deviation of the sampling distribution, we need the sample statistic for multiple samples. However, in reality we typically only have one sample! How do we know how much sample statistics vary, if we only have one sample? Calculating the Standard Error Standard Deviation: Measures the spread of the distribution of values. (e.g., the distribution of sample values for variable x). Standard Error: Measures the standard deviation of the sampling distribution (i.e., the distribution of sample statistics). Margin of Error: The amount added and subtracted to a point estimate to calculate a confidence interval for a population parameter. Terms