PY 427 Statistics 1Fall 2006 Kin Ching Kong, Ph.D Lecture 6 Chicago School of Professional Psychology.
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PY 427 Statistics 1Fall 2006 Kin Ching Kong, Ph.D Lecture 6 Chicago School of Professional Psychology
Agenda Hypothesis Testing ( review & continue) The 4 Steps of Hypothesis Testing Directional (One-Tailed) Tests The z-Score as a ratio: obtained difference difference due to chance Errors in Hypothesis Testing Type I Errors Type II Errors Assumptions for Hypothesis Tests with z-Scores Random Sampling Independent Observations is unchanged by the treatment Sampling distribution is normal Effect Size and Power
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 noted that stimulation during infancy can have profound effects on the development of infant rats. A researcher would like to know 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. e.g. =.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 = + 1.96 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 = -1.00 Since z-Score is within + 1.96, fail to reject H 0
Directional (One-Tailed) Tests Directional hypothesis test: the hypotheses (H 0 and H 1 ) specify the direction of the effect, that is, either an increase or a decrease in the population mean. An Example The weight of 2-year-olds in the U.S. is normal with a mean of 26 pounds, and a standard deviation of 4. A researcher randomly selected a sample of 4 newborn and instructs the parents to provide extra handling. The researcher predict that the extra handling will produce an increase in weight at age 2. Sample mean weight at age 2 = 29.5 pounds.
Directional Tests, An Example Step 1: State the Hypotheses H 0 : < 26 (there is no increase in weight) H 1 : > 26 (there is an increase in weight) Step 2: Define the Critical Region For =.05, z = +1.65 (or +1.64) Step 3: Compute Test Statistic z = M – M = / = 4/2 = 2 M z = (29.5 – 26)/2 = 3.5/2 = 1.75 Step 4: Make a Decision Since 1.75 > 1.65, that is, the sample mean is in the critical region, we reject H 0 and conclude that extra handling does result in increase growth for infants.
z-Score as a Ratio of Differences z = M - M = sample mean – hypothesized population mean standard error between M and z = obtained difference difference due to chance e.g. a z-Score of 3 means that the obtained difference between sample and hypothesis is 3 times bigger than expected by chance.
Errors in Hypothesis Testing Hypothesis testing is an inferential process that uses limited information (sample) to make inference about the whole population. There is always a possibility that an incorrect conclusion has been made. Two Types of Error in Hypothesis Testing: Type I Error: A null hypothesis that is actually true is rejected and the alternative hypothesis accepted. The hypothesis test is structured to control the risk of committing a Type I error. The alpha level is the probability that the test will lead to a Type I error. That is, the probability of obtaining sample data in the critical region when the null hypothesis is true. Type II Error: Fail to reject a null hypothesis that is really false. Type II Error depends on a variety of factors. Type II Error is represented by Table 8.1 of your book
Selecting an Alpha Level Alpha level ( ) serves 2 functions: Define the critical region by defining what’s “very unlikely” outcomes Determines the probability of a Type I error Selecting an alpha level ( ) Minimize the risk of a Type I error But lower would push the critical region farther out, making it harder to reject the H 0, increasing Type II error
Assumptions for Hypothesis Tests with z-Scores Random Sampling: ensure that the sample is representative of the population. Independent Observations: each observation is not influenced by any other observations. is unchanged by treatment (homogeneity of variances), that is, the effect of the treatment is to add (or subtract) a constant amount to every score. Normal Sampling Distribution: we use the unit normal table to identify the critical value.
Effect Size A hypothesis test tells you whether there is a treatment effect, but it does not tell you the magnitude of the effect. Statistical significant effect vs. substantial effect Measuring Effect Size: Cohen’s d = mean difference standard deviation So the effect size is standardized by , i.e., measured in unit of Figure 8.8 of your book Small effect: d < 0.2 Medium effect: d around 0.5 Large effect: d > 0.8
Power Power: the probability that the test will correctly reject a false H 0. That is, power is the probability that the test will identify a treatment effect when one really exists. Power = 1 – Some factors that affect power: Effect size: as effect size increase, power increase. Fig.8.9 of your book Fig.8.9 of your book Sample size: as sample size increase, power increase. Population variance: as population variance increase, power decrease. Alpha level: as decrease, power decrease.
An Example Scores on a standardized test are normally distributed with = 65, =15. A sample of 25 individuals were randomly selected and given special training. The average test score for this sample is 70. Is there evidence that the training has an effect on test scores? Use =.05 Answer