PY 427 Statistics 1Fall 2006 Kin Ching Kong, Ph.D Lecture 5 Chicago School of Professional Psychology

Agenda Samples & Sampling Distributions Samples & Sampling Error Sampling Distributions Distribution of Sample Means Central Limit Theorem The Expected Value of M The Standard Error of M Probability & The Distribution of Sample Means z-Score for Sample Means Introduction to Hypothesis Testing

Preview of Inferential Statistics Experiment at the end of Ch. 5 The distribution of adult rat weights is normal with a  of 400g, and a  of 20g. A researcher selects one newborn rat and injects it with a growth hormone. When this rat reaches maturity, it’s weight is obtained. Let say this rat weights 450g. Did the treatment (hormone) have an effect on adult weight? Another way to ask this question: Did the treated sample (this rat of 450g) came from the original population or a new population of hormone injected rats?

Samples & Sampling Error Most research studies has sample size >1 Each sample can be summarized by a typical value, a statistic (the sample mean M) z-Scores and Probability can be applied to sample means, M’s, similar to how they are applied to raw scores, X’s. Sampling Error: the discrepancy, or amount of error, between a sample statistic and its corresponding population parameter Sampling Error is to sample means as individual differences are to raw scores

Sampling Distribution Distribution of Sample Means: is the collection of sample means for all the possible random sample of a particular size (n) that can be obtained from a population. Sampling Distribution: a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.

Demo of a sampling distribution of M Example 7.1 of your book A population of 4 scores: 2, 4, 6, 8 Distribution of sample means for n = 2 Table 7.1 Characteristics of the Sampling Distribution of M: The sample means tends to pile up around the population mean,  = 5 The distribution of M is approximately normal The distribution of M can be used to answer probability questions re: sample means. e.g. p(M > 7) =?

Central Limit Theorem Central Limit Theorem: For any population with mean  and standard deviation , the distribution of sample means for sample size n will have a mean of  and a standard deviation of  / and will approach a normal distribution as n approaches infinity. The distribution of sample means will be almost perfectly normal if either: The population from which the sample are selected is a normal distribution. The sample size, n, is relatively large, >30

The Distribution of Sample Means The mean of the distribution of sample means is equal to the population mean, , and is call the Expected Value of M. The standard deviation of the distribution of sample means is call the Standard Error of M,  M.  M measures the standard amount of difference between M and  due to chance.

Standard Error of M Standard Error =  M =  / Sample size: the larger the n, the smaller  M The law of large numbers: the larger the sample size, the more probable the sample mean will be close to the population mean. Population standard deviation: the larger , the larger  M Expressed in terms of the variance:  M =

Probability & Sample Means The sampling distribution of means, M, can be used to find the probability of any M similar to how the distribution of raw scores, X, can be used to find the probability of any X. z = M -   M Exercises: The distribution of SAT scores is normal with  = 500 and  = 100. If we take a random sample of 25 students, what is the probability that the sample mean will be greater then 540? Answer Answer What range of values do we expect the mean for a sample of 25 students to be 80% of the time?AnswerAnswer

In the Literature Reporting Mean & Standard Deviation M = meanSD = sample standard deviation Reporting Mean & Standard Error M = meanSE or SEM = standard error of M Graph Figure 7.6Figure 7.7 Figure 7.6Figure 7.7

Preview of Inferential Statistics The Rat Study: The distribution of adult rat weights is normal with a  of 400g, and a  of 20g. A researcher selects n=25 newborn rats and injects them with a growth hormone. When this sample reach maturity, their weights are obtained. The Logic of Inferential Statistics: To make a decision of whether the hormone had an effect, the researcher compares the sample of treated rats to samples of untreated rats (using the distribution of sample means) If the treated sample is noticeably different from the untreated samples, then there is evidence that the hormone had an effect. If the treated sample looks like what is expected for a sample of untreated rats, then there is no evidence that the hormone had an effect. Figure 7.9 of your book

Introduction of Hypothesis Testing Hypothesis testing is a formalized procedure that uses sample data to evaluate a hypothesis about a population parameter. Four Steps of Hypothesis Testing Step 1: State the Hypotheses H 0 :  = 400g H 1 :  = 400g Step 2: Set the Criteria for a Decision Deciding what is noticeably different, selecting a p, alpha or significance level. Step 3: Compute Sample Statistics z-Score Step 4: Make a Decision reject or fail to reject H 0

Illustration of the Steps of Hypothesis Testing Example 8.1 of your book Psychologists have noted that stimulation during infancy can have profound effects on the development of infant rats. A researcher would like to determine whether stimulation during infancy has an effect on human development. From national health statistics: the mean weight for 2- year-old children is  = 26 pound. The distribution is normal with  = 4 pounds. A sample of n = 16 newborn infants are selected and their parents given instructions to increase handling and stimulation of the infants. At age 2, the children’s weight are obtained.

Step 1: State the Hypotheses Hypotheses are statements about population parameters. The null hypothesis (H 0 ) usually states that there is no effect (no treatment effects, no relationship between variables, no changes etc). H 0 :  infants handled = 26 pound The alternative hypothesis (H 1 ) states that there is a change, a difference, an effect or a relationship between the independent and dependent variables. H 1 :  infants handled = 26 pound

Step 2: Set the Criteria for a Decision The alpha (  ) level or the level of significance is a probability value that is used to defined the very unlikely sample outcomes if the null hypothesis is true.  =.05 The critical region is composed of extreme sample values that are very unlikely to be obtained if the null hypothesis is true. The boundaries for the critical region are determined by the alpha level. If sample data fall in the critical region, the null hypothesis is rejected. The critical region: find z-Scores for the extreme 5% z = Figure 8.3 of your book

Step 3: Compute Sample Statistics z = M –   M  M =  / Suppose the sample of n= 16 infants produced a sample mean of M = 30 at age 2  M = 4/ = 4/4 =1 z = (30 – 26)/1 = 4.00

Step 4: Make a Decision Since z-Score for this sample is 4.00, which is beyond the boundary of 1.96, the sample z- Score is in the critical region. The null hypothesis is rejected, and the researcher can conclude that there is evidence that the increased handling had an effect on the infants weight. Suppose the sample mean = 25 at age 2  M = 4/ = 4/4 =1 z = (25 – 26)/1 = Since z-Score is within , fail to reject H 0