An Introduction to Making Inferences Chapter 10 Reading Assignment pp

Inferential Statistics Inferential Statistics –Utilizes sample data to make estimates, decisions, predictions or other generalizations about the larger set of data(i.e., infer), i.e., numbers that help to assess the likelihood that randomly drawn samples will display patterns that reflect the patterns in the larger population

Terminology 3 Recall: Population: a set of elements--usually people, objects, transactions or events—about which information is desired. Sample: a smaller, more manageable set of elements—a subset of a population—selected to represent the population for which it is drawn Mean is the most stable of the three measures, i.e., if several random samples are taken from the same data set, there will be less variability among the means than among the medians or modes.

Sampling 4 Question: How likely is it that the sample selected is representative of the population? Probability Sample: a sample selected in such a way that each element in the population has the same chance of being drawn into the sample as every other element in the population Before drawing samples, come up with sampling frame, or a list of elements in the population P. 407—random is not necessarily representative

Normal Distribution and z- score 5 When the shape of the histogram of the distribution is perfectly bell-shaped (unimodal and symmetrical—single mode and as many cases above the mean as below) Skills 1, p. 408 Z-score: an expression of the relationship between a particular value in a distribution and the mean in units of the standard deviation Formula: Z=(X_i-mean)/(s) (s=std dev) Example, p. 409 Interpretation, p. 410; Skills 2, p. 410

Z-score 5 Reading Tables—Appendix A Skills 3, p. 410 Z scores in SPSS—follow pp Skills 4, p. 414 JTS: 22 years educ (full-time): Excel—stats z

Sampling Statistics 4 Sampling statistic: any statistic that describes the distribution of values for a variable, or the relationships between variables, in a sample Eg./ Z scores, measures of central tendency, measures of dispersion, and measures of association Sampling distribution: the distribution of all possible samples of a given size from a particular population P. 415—20 people in class; p. 416— table 10.2

Sampling Distribution of sample means 4 Hypothetical distribution of all possible sample means of a given sample size from a particular population Formula 10.2 p. 417; example; Skills 5 Standard deviation of the sample means: tells if homogeneous or heterogeneous Example p.418; Skills 6

Z scores for a population 7 Formula 10.4 Example, p. 420 Skills 7 (computations for variables on p. 417) Std dev of sampling distribution of sample means—also called the standard error of the mean Formula 10.5; example; Avoiding pitfalls, p. 422 Skills 8 Hypothetical distribution of all possible sample means of a given sample size from a particular population Formula 10.2 p. 417; example; Skills 5 Standard deviation of the sample means: tells if homogeneous or heterogeneous Example p.418; Skills 6

Central Limit Theorem 2 Note that the sampling distribution of the sample mean is approximately normal for large sample sizes CLT: If a random sample of n observations is selected from a population (any population), then when n is sufficiently large, the sampling distribution of the sample mean will be a normal distribution. The larger the sample size, the better will be the normal approximation to the sample distribution of the sample mean.

Central Limit Theorem-- consequences 2 Computing the mean and median for all possible samples in a sampling distribution: The mean and the median for the sampling distribution of sample means should be identical (to the properties of a normal distribution) or fairly close to it There should be only one mode The distribution of the means should be symmetrical CLT demo: /CLT/clt.html

Homework P.429/14,15,18,19,20 Spss/ 1 Hand in: gen ex # 13, SPSS #3