Chapter 10: Introduction to Statistical Inference

Recall that the purpose of descriptive statistics is to make the collected data more easily comprehensible and understandable. Some tools we’ve examined in descriptive statistics include frequency distributions, measures of central tendency, and measure of dispersion. Because it is not always possible to address every member of the population, we take samples. The statistical question that needs to be answered is whether or not the characteristics observed in the sample are likely to reflect the true characteristics of the larger population from which the sample was taken. Inferential statistics provide us with the tools we need to answer this question.

In inferential statistics, the goal is to make statements about the characteristics of a population based on what we have learned from the sample data. Inferential statistics has two broad applications: estimation and hypothesis testing. Estimation uses information contained in a sample to make a “guess” of the population value. Hypothesis testing determines whether or not a hypothesized value or relationship in the population is likely to be true.

Recall that in a random sample, every member of the population has an equal chance of being selected. Also recall that a descriptive measure calculated from a sample is statistic. We use statistics as a way to estimate a population parameter. Just how accurately does a sample statistic estimate a population parameter???

Typically we usually draw only one sample from a population and use that sample statistic calculated as an estimate of the population parameter. If we drew a different sample, our estimate for the population would be slightly different. So if we calculated a mean, we would end up with a slightly different mean. If we took 6 samples from the same population, we would likely have 6 different means. A sampling distribution is the distribution of numbers, obtained by calculating a sample statistic, for all possible samples of a given size drawn from the same population.

Let’s say we did have a population and pulled six samples. The means of those six samples are 20, 23, 24, 21, 22, and 25. Which one would you report? You might want to report the mean of those sample means.

A sample statistic will not always equal the population parameter (in most cases it won’t). Random sampling error is the measure of the extent to which the sample statistic differs from the population parameter, due to random chance. Parameter = statistic + random sampling error Note: Random sampling error and standard error are interchangeable terms.

Sample Size and Standard Error As sample size increases, standard error decreases.

The Central Limit Theorem Just how large does n have to be? The rule of thumb is that n has to be 30 or more. Once we know we are dealing with a Normal distribution, we can utilize the Empirical Rule and the standard Normal table to help us attain information about our population.

Back to Z-Scores When dealing with a sample mean, calculate its z-score by: When the population standard deviation is unknown: These z-scores will measure how many standard deviations the sample mean deviates from the population mean.

Example: Calculate and interpret the z-score for the following data. Interpretation: The sample mean of 50 is 2 standard deviations above the population mean.

Example: Calculate and interpret the z-score for the following data. Interpretation: The sample mean of 85 is 1.5 standard deviations below the population mean.

=.3917

.9582