Understanding Inference: Confidence Intervals I. Questions about the Assignment. The Big Picture. Statistic vs. Parameter. Statistic vs.

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Julian Jennings
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1 Questions about the Assignment If your answer is wrong, but you show your work you can get more partial credit. Understanding Inference: Confidence Intervals I parameter versus sample statistic Uncertainty in estimates Sampling distribution Confidence interval The Big Picture Statistic vs. Parameter Statistical Inference Sampling Sample 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. The Big Picture Statistic vs. Parameter Parameter Statistical Inference Sampling Sample Statistic Mean Sample Statistics Parameters μ (mu) Proportion p Std. Deviation s (sigma) Correlation r ρ (rho) Slope b β (beta) 1

2 Obama s Approval Rating Gallup surveyed 1,500 Americans between June 9 th -11 th 2012 and 49% of these people approved of the job Barack Obama is doing as president. What is the population? ~330million (All Americans) What is the sample size? 1,500 Is this categorical or quantitative variable? Categorical For categorical variables, what sample statistic are we interested in? Sample proportion Sample statistic: (sample proportion) =.49 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? parameter: (population proportion) p =? Point and Interval Estimates 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??? Obama s Approval Rating Point Estimate: =.49 Interval Estimate: point estimate margin of error = (0.46, 0.52) Between 46% and 52% of Americans currently approve of the job Obama is doing as president. Important Points The population parameter is a fixed value. Sample statistics vary from sample to sample. They will not match the population parameter exactly. Reese s Pieces What proportion of Reese s pieces are orange? 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! 2

3 Let s Run Our Own Study When conducting a study, we need to select a sample size. Typically, we take only one sample, but because we re interested in knowing how much our sample statistic varies from sample to sample, we ll take multiple samples. Each person take a random sample of 10 Reese s pieces. Let s Run Our Own Study What is our population? 1,500 What is our sample size? 10 How many samples did we take? 6 What is your sample distribution (i.e. orange vs. not orange)? What is your sample proportion? What is the range of plausible values for the population proportion? What is the mean proportion of the sampling distribution? The sampling distribution will be centered around the true population parameter. Sampling Distribution A sampling distribution is the distribution of sample statistics computed from different samples of the same size taken from the same population. In the Reese s pieces sampling distribution, what does each dot represent? A. One Reese s piece B. One sample statistic The sampling distribution shows us how the sample statistic varies from sample to sample. Sampling Distribution: Shape and Center If samples are randomly selected and the sample size is large enough, the sampling distribution will be symmetric and bell-shaped. centered at the value of the population parameter. The sampling distribution is different from the sample distribution. The sample distribution is the distribution of values for variable x collected from one sample. The sampling distribution is the distribution of sample statistics collected from multiple samples. Sampling Distribution: Spread To assess the accuracy of our point estimate, we need to know how much the sample statistic varies from sample to sample. (i.e., we need to know the spread of the sampling distribution.) In the Reese s pieces sampling distribution we generated, what is the range of plausible values for the population proportion? We use the spread of the sampling distribution to determine the margin of error for a statistic. What is a standardized way to measure the spread of a distribution? Sampling Distribution: Standard Deviation Calculate the standard deviation of the sample statistics in the sampling distribution. s n x 2 i x i 1 n 1 n = the number of samples taken x i = the sample statistic for sample i = mean value for all of the sample statistics As the standard deviation (i.e., spread) of the sampling distribution decreases, the margin of error will decrease. As the variability (i.e., spread) in the sampling distribution decreases, the uncertainty in the estimate decreases. 3

4 The Importance of Sample Size 3 Sampling Distributions n = 1,000 n = 200 n = 50 Each dot represents a sample statistic. The number of samples taken to generate these sampling distributions is the same. What varies for each sampling distribution is the size of the sample taken to calculate the sample statistic. The sample size does not affect the shape of the sampling distribution. The sample size does not affect the center of the sampling distribution. The sample size does affect the spread of the sampling distribution. As the sample size increases, the spread decreases. Sample Size The sample size influences the spread of the sampling distribution (i.e., the variation in sample statistics), which influences the margin of error for our estimate of the population parameter. If we increased the sample size to 100, the standard deviation of the sampling distribution will... A. increase For each sample, the sample statistic (i.e., the proportion of orange pieces) would be closer B. decrease to the proportion of the population and thus C. remain the same closer to each other. and the margin of error for our point estimate will A. increase B. decrease C. remains the same Hypothesis Increasing the sample size will cause the standard deviation of the sampling distribution to decrease. Let s Test Our Hypothesis Random Samples If you take random samples, the sampling distribution will be centered around the true population parameter. If sampling bias exists (if you do not take random samples), the sampling distribution may provide inaccurate information about the true population parameter. Confidence Intervals Confidence Intervals A confidence interval for a population parameter estimate is an interval computed from sample data that will contain the true population parameter for a specified proportion of all samples. The confidence level is the proportion of samples whose intervals contain the true population parameter. The confidence level indicates how confident we are that our interval contains the population parameter. A 95% confidence interval will contain the true population parameter for 95% of all samples. We are 95% confident that the true population parameters falls within this range. Sampling Distribution Parameter % Proportion Confidence Interval p Sample Statistic >2 SDs *The standard deviation used to calculate the confidence interval is the standard deviation of the sampling distribution (not the sample distribution). The population parameter ( ) is fixed. It is typically not known. The sample statistic (x i ) is random. It depends on the sample. The confidence interval (x i 2SD)* is random. It depends on the sample statistic. The sampling distribution is comprised of the sample statistics and is centered on the population parameter. 95% of the sample statistics will fall within 2 standard deviations of the population parameter. 95% of the sample intervals will contain the population parameter. 4

Confidence Intervals A 95% confidence interval can be created by: sample statistic 2 standard deviations point estimate margin of error Standard Error: The Standard Deviation of the Sampling

This is done to clearly distinguish it from the standard deviation of the sample distribution. The point estimate is calculated from our sample.

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.

Part I: Graded Problems 3.12, 3.16, 3.24, and 3.54 Assignment Part II: (Type up this assignment in a Word document) Goto http://sda.berkeley.edu/cgi-bin/hsda?

5 Confidence Intervals A 95% confidence interval can be created by: sample statistic 2 standard deviations point estimate margin of error Standard Error: The Standard Deviation of the Sampling Distribution 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. The point estimate is calculated from our sample. The margin of error is calculated from the sampling distribution. Summary 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? Part I: Graded Problems 3.12, 3.16, 3.24, and 3.54 Assignment Part II: (Type up this assignment in a Word document) Goto 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 Calculating Correlations from the GSS 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. This is what will pop up in the new window. This is the correlation (r) score for the two variables 5

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 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 Calculating the Standard Error 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? Terms 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. 6

Business Statistics. Lecture 10: Course Review

Business Statistics. Lecture 10: Course Review Business Statistics Lecture 10: Course Review 1 Descriptive Statistics for Continuous Data Numerical Summaries Location: mean, median Spread or variability: variance, standard deviation, range, percentiles,