Page Title PSY 305 Module 3 Introduction to Hypothesis Testing Z-tests Five steps in hypothesis testing State the research and null hypothesis Determine characteristics of comparison distribution
Five steps in hypothesis testing Determine the cut-off score at which null will be rejected Calculate statistic Make decision Normal Distribution Properties we know everything about this distribution! Mean = 0; Standard Deviation = 1 Area under the curve always equals 1.0, and we know all proportions under the curve Normal Distribution Bell-shaped Symmetrical The upper half is a mirror image of the lower half Values of the mean, median, and mode are the same
Normal Distribution Each point along the x-axis corresponds with something called a z-score. We can make our scores (or observations) map onto this normal curve by transforming them into z-scores Normal Distribution Z scores IQ scores Let s say you have an IQ score of 132 Is this good or bad? Reference group
Z scores A z-score indicates how many standard deviations an observation is above or below the mean. Also called a standard score Z Tests A z-score simply compares one score to the population. A z-test actually compares a sample mean to the population. Calculating Z Tests X z X z = the z-score you re calculating X = sample mean µ = mean σ X
Standard error of the mean The standard deviation of the sampling distribution of means is called the standard error of the mean. The formula for the true standard error of the mean is X X N Let s practice Let s practice using our IQ example Mean IQ = 132 µ = 100, σ = 16 n = 10 What is the z obtained? 6.324 Your turn The sample mean is 95 The population mean is 90 with a standard deviation of 10. n = 8 What is the z obtained?
Answer 95-90 = 5 Standard error of mean 10/ 8 = 3.536 5/3.536 = 1.41 Z obtained = 1.41 Evaluating the tail of the distribution Decision Rules Assess the null hypothesis Can directly test the probability of chance events Cannot test the probability of the alternative hypothesis Decision rules If the obtained probability is equal to or less than a critical value, we reject the null. The critical value is called the alpha (α) level. Indirectly accept the alternative hypothesis. Reject Ho, say results are significant
Decision rules If the obtained probability α, reject H 0 If the obtained probability > α, fail to reject H 0 or retain H 0. In psychology, we often use α =.05 or α =.01 Decision rules So, if we set α =.05, we are willing to limit the chance of rejecting the null when it is true to 5 times out of 100. Decrease our chances of making Type 1 error. Correct Decisions, Type I, and Type II errors When making a decision, four possible outcomes Correct Decision (Two Types) You said there was no effect and you were right. You said there was an effect and you were right.
Type I and Type II errors Type I Error You said there was a significant effect when there really wasn t. You rejected the null hypothesis when you should have retained it. Type I and Type II errors Type II Error you said there was no effect when there really was. You retained the null hypothesis when you should have rejected it. Evaluating Tale of the Distribution We test the tail of the distribution beginning with the obtained results If nondirectional- evaluate both tails. If directional- evaluate only the tail that is in the direction of alternative hypothesis.
One and Two Tailed Tests Must decide if test is one or two tailed before setting alpha level. Always use a two-tailed test unless we plan to retain the null hypo when results are extreme in the direction opposite to the predicted direction Two tailed probability Outcomes we evaluate occur under both tails of the distribution Set.05 as our alpha but we have to divide it between the two tails. So, with 5% significance, are actually looking at 2.5% at each tail Two tailed probability Our cut-off at the 5% level is -1.96 and +1.96 Our cut-off at the 1% level is -2.58 and +2.58 These will always be used as the cut-off z-scores so be sure you learn these values!!
Two tailed probability.025.025 One tailed probability All outcomes are under one tail of the distribution So, set alpha at.05 For 5% chance, the cut-off is +1.64 or -1.64 For 1% chance, the cut-off is +2.33 or -2.33
Critical Values Where do these cut-off values come from? Our z-tables in the back of the book Look at column C to find percentage of scores beyond z This tells us which z-score we need to use for our cut-off Previous Examples Recall example where Mean IQ = 132 µ = 100, σ = 16 n = 10 z obtained = 6.324 Is this z significant at p <.05 for a two- tailed test? Previous Examples z obtained = 6.324 z critical for p <.05, two tailed = +/-1.96 6.325 > 1.96 Yes, the sample s IQ is different from the population.
Previous Examples Recall example where Your sample mean = 95 The population mean is 90 with a standard deviation of 10. n = 8 Z obtained = 1.41 Is this z significant? Previous Examples z obtained = 1.41 z critical for p <.05, two tailed = +/-1.96 1.41 < 1.96 No, the sample is not significantly different from the population Note Draw your normal distribution. Mark the critical region and cut-off values. Determine if the z is significant.
Statistical significance Does not tell us if effect is important Effect size tells us this Does not mean the effect will replicate Does not mean effect will generalize to other populations Statistical significance It tells you that difference is large enough that it would not occur by chance more than some probability We usually set p <.05 A difference that large will occur 5% of the time