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.
Statistical Estimation
Statistics – observed values; computed based on the sample data
Parameters – estimated values; estimated based on sample statistics
Example:
Motivation Scale: N = 400 Sample Mean = 80
What is the population mean on this motivation scale?
In this example, the same mean is 80. It is not likely that the population mean would be the same as the sample mean since it is a different set of individuals. In order to estimate the population mean, the standard deviation of this variable in the population must be known.
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.
Confidence Intervals
The formulas for the confidence interval and margin of error can be combined into one formula.
The Reasoning of Statistical Estimation
Where does the formula for computing a confidence interval come from? The true value of the population mean is never known – it can only be approximated or estimated. The best way to do this is to select a large number of random samples of the same size from the population. The mean from each random sample will be slightly different. The average of these sample means is the population mean.
For instance, the mean for the sample in the example was 80, but if another sample was selected the mean might be 78 or 83. If a large number of sample means were represented graphically, they would have a Normal distribution. The mean of this distribution is the same as the sample mean, but the standard deviation of this distribution is equal to the standard deviation of the variable in the population divided by the square root of the sample size. This is the reason that the standard deviation is divided by the square root of n in the formula, instead of the simple standard deviation, because this formula represents the standard deviation of the distribution of many sample means. When working with real data it may not be feasible to select a very large number of random samples, but if researchers were able to do so, the samples would form a Normal distribution.
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.”
