# Hypothesis Testing. Reminder of Inferential Statistics. Hypothesis Testing: Introduction

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1 Hypothesis Testing PSY 360 Introduction to Statistics for the Behavioral Sciences Reminder of Inferential Statistics All inferential statistics have the following in common: Use of some descriptive statistic Use of probability Potential for estimation Sampling variability Sampling distributions Use of a theoretical distribution Two hypotheses, two decisions, & two types of error Hypothesis Testing: Introduction This is the last of the seven topics common to all inferential statistics, and so it integrates all of the other six. Principally, hypothesis testing uses probability and the sampling distribution of a statistic to make decisions about a parameter Hypothesis testing is the process of testing tentative guesses about relationships between variables in populations. These relationships between variables are evidenced in a statement, a hypothesis, about a population parameter 1

2 Hypothesis Testing: Examples Rat-shipment Are the rats defective? Or are they OK? If µ=33 and σ 2 =361, is the X=44.4 from the sample of N=25 rats significantly different from 33? Compute z X =(X- µ)/ (σ 2 /N)= ( )/ (361/25)= 11.4/3.8=3.00 Find p(x>44.4)=p(z>3)= It is unlikely the rats came from a population with µ=33 for the mean run time. So we decide that µ 33 and that the rats are defective Hypothesis Testing: Examples IQ of deaf children are the deaf children lower in IQ? Or are they average? If µ=100 and σ 2 =225, is the X=88.07 from the sample of N=59 deaf children significantly lower than 100? Compute z X =(X- µ)/ (σ 2 /N)= ( )/ (225/59) = /1.95=-6.11 Find p(x<88.07)=p(z<-6.11)< It is unlikely the deaf children came from a population with µ=100 for the mean IQ. So we decide that µ<100 and that the deaf children have lower IQ scores Remember, this is due to the fact that their language, ASL, is not English, so they score lower on the verbal part of the total IQ test Hypothesis Testing: Key Terms Test statistic: a statistic used only for the purpose of testing hypotheses; e.g., z X Assumptions: conditions placed on a test statistic necessary for its valid use in hypothesis testing; for z X, the assumptions are that the population is normal in shape and that the observations are independent Null hypothesis: the hypothesis that we test; Alternative hypothesis: where we put what we believe; H 1 Both and H 1 are stated in terms of a parameter 2

3 Hypothesis Testing: Key Terms Significance level: the standard for what we mean by a small probability in hypothesis testing; α Directional and non-directional hypotheses One- and two-tailed tests, critical, and rejection Decision rules: decision rules p-value decision rules H O and H 1 Rat-shipment We start with H 1. We believe that there is something wrong with the rats, or that µ 33. So we have H 1 : µ 33 Next, we state. The null is always the opposite of the alternative. Within and H 1, the set of potential of the parameter to be tested usually contains all possible numbers. The null hypothesis usually has the equals in it. So we have : µ=33 IQ of deaf children Again, we start with H 1. We believe that the deaf children will score lower on the IQ test because English is not their native language, or that µ<100. So we have H 1 : µ<100 Next, we state. So we have : µ>100 Significance Level The significance level is the small probability used in hypothesis testing to determine an unusual event that leads you to reject The significance level is symbolized by α (alpha) The value of α is almost always set at The value of α is chosen before data are collected If is rejected when, here are examples of what you say: The mean of the IQ of deaf children, X=88.07, is significantly lower than 100, z=-6.11, p<.00003, one-tailed test The mean of the run times, X=44.4, is significantly different from 33, z=3.00, p=

4 Directional and Non-Directional Hypotheses Directional hypotheses specify a particular direction for of the parameter IQ of deaf children : µ>100, H 1 : µ<100 Non-directional hypotheses do not specify a particular direction for of the parameter Rat shipment : µ=33, H 1 : µ 33 Another Suppose you believe that dancers are more introverted than other people. You have N=26 dancers and know that for this age group with your male/female ratio that µ=19.15 for introversion So you have : µ<19.15 and H 1 : µ>19.15 One- and Two-Tailed Tests, Critical Values, and Values One- and two-tailed tests: A one-tailed test is a statistical test that uses only one tail of the sampling distribution of the test statistic A two-tailed test is a statistical test that uses two tails of the sampling distribution of the test statistic Critical are of the test statistic that cut off α or α/2 in the tail(s) of the theoretical reference distribution are the of the test statistic that lead to rejection of One- and Two-Tailed Tests, Critical Values, and Values Rat shipment : µ=33, H 1 : µ 33 Two-tailed test Critical are =-1.96 and = 1.96 are <-1.96 and > Critical 4

5 One- and Two-Tailed Tests, Critical Values, and Values IQ of deaf children : µ>100, H 1 : µ<100. One-tailed test is = are < One- and Two-Tailed Tests, Critical Values, and Values Introversion of dancers : µ<19.15, H 1 : µ>19.15 One-tailed test is =1.645 are > Critical Value Decision Rules Rat shipment : µ=33, H 1 : µ 33 Reject if the observed z X < or if z X >1.96 The observed z X was z X =3.00 Reject Ho: µ=33 because 3.00>1.96 (observed z X > ) Critical Z=3.00 5

6 Critical Value Decision Rules IQ of deaf children : µ>100, H 1 : µ<100 Reject if the observed z X < The observed z X was z X =-6.11 Z= Reject : µ>100 because -6.11< (observed z X < ). Compute z X for Introversion of Dancers Remember, you believe that dancers are more introverted than other people. You have N=26 dancers and know that for this age group with your male/female ratio that µ= So you have : µ<19.15 and H 1 : µ> Introversion of dancers are the dancers higher in introversion? Or are they average? If µ=19.15 and σ 2 = , is the X=19.79 from the sample of N=26 dancers significantly higher than 19.15? Compute z X =(X- µ)/ (σ 2 /N)= ( )/ ( /26)=.64/.85=.76. Critical Value Decision Rules Introversion of dancers : µ<19.15, H 1 : µ>19.15 Reject if the observed z X >1.645 The observed z X was z X =.76 Retain Ho: µ<19.15 because.76<1.645 (observed z X < ) Z=.76 6

7 p-value Decision Rules Rat shipment : µ=33, H 1 : µ 33 Reject if the SPSS (2-tailed) p- value is < The SPSS p-value is.0026 Reject : µ=33 because.0026< (p-value is < α) Critical Z=3.00 p-value Decision Rules IQ of deaf children: Ho:µ>100, H1: µ<100 SPSS p= Reject if ½ the SPSS p-value <α, and the observed z X is in the tail specified by H 1 ½ the SPSS p-value is and the observed z X was in the left tail (as in H 1 ) So, reject Ho: µ>100 Z= p-value Decision Rules Introversion of dancers Ho: µ<19.15, H1: µ>19.15 SPSS p=.4472 Reject if ½ the SPSS p-value <α, and the observed z X is in the tail specified by H 1 ½ the SPSS p-value is.2236 and the observed z X was in the right tail (as in H 1 ) So, retain : µ< Z=.76 7

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