Understanding Confidence Intervals for Population Means, Study notes of Statistics

Download Understanding Confidence Intervals for Population Means and more Study notes Statistics in PDF only on Docsity! Confidence Intervals Diana Mindrila, Ph.D. Phoebe Balentyne, M.Ed. Based on Chapter 14 of The Basic Practice of Statistics (6th ed.) Concepts:  The Reasoning of Statistical Estimation  Margin of Error and Confidence Level  Confidence Intervals for a Population Mean  How Confidence Intervals Behave Objectives:  Define statistical inference.  Describe the reasoning of statistical estimation.  Describe the parts of a confidence interval.  Interpret a confidence level.  Construct and interpret a confidence interval for the mean of a Normal population.  Describe how confidence intervals behave. References: Moore, D. S., Notz, W. I, & Flinger, M. A. (2013). The basic practice of statistics (6th ed.). New York, NY: W. H. Freeman and Company. Statistical Inference  The purpose of collecting data on a sample is not simply to have data on that sample. Researchers take the sample in order to infer from that data some conclusion about the wider population represented by the sample.  These notes will cover how to estimate the mean of a variable for the entire population after computing the mean for a specific sample.  For example, a researcher is interested in estimating the achievement motivation of first year college students. The researcher must select a random sample of students, administer a motivation scale, and then compute the average score for the entire sample. Based on this average score, he or she can then make an inference about the motivation of the entire population of first year college students. Statistical Inference Statistical inference provides methods for drawing conclusions about a population from sample data. Estimating the Population Mean Confidence Level The confidence level is the overall capture rate if the method is used many times. The sample mean will vary from sample to sample, but the method estimate ± margin of error is used to get an interval based on each sample. C% of these intervals capture the unknown population mean 𝜇. In other words, the actual mean will be located within the interval C% of the time. Confidence interval = sample mean ± margin of error  The population mean for a certain variable is estimated by computing a confidence interval for that mean.  If several random samples were collected, the mean for that variable would be slightly different from one sample to another. Therefore, when researchers estimate population means, instead of providing only one value, they specify a range of values (or an interval) within which this mean is likely to be located.  To obtain this confidence interval, add and subtract the margin of error from the sample mean. This result is the upper limit and the lower limit of the confidence interval. The confidence interval may be wider or narrower depending on the degree of certainty, or estimation precision, that is required. Margin of Error  In order to find a confidence interval, the margin of error must be known.  The margin of error depends on the degree of confidence that is required for the estimation.  Typically degrees of confidence vary between 90% and 99.9%, but it is up to the researcher to decide.  The level of confidence is represented by z* (called z star).  It is also necessary to know the standard deviation of the variable in the population. (Note: the population standard deviation is NOT the same as the sample standard deviation).  Finally, the size of the sample n will be used to compute the margin of error.  In the above example, a confidence level of 95% was selected. The value of z* for a specific confidence level is found using a table in the back of a statistics textbook. The value of z* for a confidence level of 95% is 1.96.  After putting the value of z*, the population standard deviation, and the sample size into the equation, a margin of error of 3.92 is found. Margin of error = z* ∙ population standard deviation √𝑛 Confidence Intervals  The formulas for the confidence interval and margin of error can be combined into one formula. How Confidence Intervals Behave The z confidence interval for the mean of a Normal population illustrates several important properties that are shared by all confidence intervals in common use.  The user chooses the confidence level and the margin of error follows.  Researchers would prefer high confidence with a small margin of error.  High confidence suggests the method almost always gives correct answers.  A small margin of error suggests the parameter has been pinned down precisely. How is a small margin of error obtained? The margin of error for the z confidence interval is: The margin of error gets smaller when: • z* gets smaller (the same as a lower confidence level C) • σ is smaller. It is easier to pin down µ when σ is smaller. • n gets larger. Since n is under the square root sign, four times as many observations are needed to cut the margin of error in half. Interpreting the Confidence Level The confidence level is the overall capture rate if the method is used many times. The sample mean will vary from sample to sample, but when the method estimate ± margin of error is used to get an interval based on each sample, C% of these intervals capture the unknown population mean μ. To say that there is 95% confidence is shorthand for “95% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter.” Confidence Intervals: The Four-Step Process State: What is the practical question that requires estimating a parameter? Plan: Identify the parameter, choose a level of confidence, and select the type of confidence interval that fits the situation. Solve: Carry out the work in two phases: 1. Check the conditions for the interval that has been chosen. 2. Calculate the confidence interval. Conclude: Return to the practical question to describe the results in this setting.