 Once you know the correlation coefficient for your sample, you might want to determine whether this correlation occurred by chance.  Or does the relationship you found in your sample really exist in the population or were your results a fluke?  Or in the case of a t-test, did the difference between the two means in your sample occurred by chance and not really exist in your population. 2

 If you set your confidence level at 0.05  Let’s assume that you collected your data with 100 different samples from the same population and calculate correlation each time. So, the maximum of 5 out of 100 samples might show a relationship when there really was no relationship (r=0) 3

 Any relationship should be assessed for its significance as well as its strength  Pearson correlation measures the strength of a relationship between two continuous variables  Correlation coefficient: r  Coefficient of determination: r 2  Significance is measured by t-test with p=0.05 (which tells how unlikely a given correlation coefficient, r, will occur given no relationship in the population)  The smaller the p-level, the more significant the relationship  The larger the correlation, the stronger the relationship 4

 You have a sample from a population  Whether you observed statistic for the sample is likely to be observed given some assumption of the corresponding population parameter. 5

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 When the test is against the null hypothesis: r xy = 0.0  What is the likelihood of drawing a sample with r xy ­ =0.0?  The sampling distribution of r is  approximately normal (but bounded at -1.0 and +1.0) when N is large  and distributes t when N is small. 7

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Quality of Marriage Quality of parent-child relationship

 Step1: a statement of the null and research hypotheses  Null hypothesis: there is no relationship between the quality of the marriage and the quality of the relationship between parents and children  Research hypothesis: (two-tailed, nondirectional) there is a relationship between the two variables 10

 CORREL() and PEARSON() 11 r=0.393

 Step2: setting the level of risk (or the level of significance or Type I error) associated with the null hypothesis  0.05 or 0.01  What does it mean?  on any test of the null hypothesis, there is a 5% (1%) chance you will reject it when the null is true when there is no group difference at all.  Why not ?  So rigorous in your rejection of false null hypothesis that you may miss a true one; such stringent Type I error rate allows for little leeway 12

 Step 3 and 4: select the appropriate test statistics  The relationship between variables, and not the difference between groups, is being examined.  Only two variables are being used  The appropriate test statistic to use is the t test for the correlation coefficient 13

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 Step5: determination of the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statistic.  From t table, the critical value=2.052 (two tailed, 0.05, df=27)  T=2.22  If obtained value>the critical value  reject null hypothesis  If obtained value<the critical value  accept null hypothesis 15

 Step6: compare the obtained value with the critical value  T Distribution Critical Values Table (Critical value r table)  compute the correlation coefficient (r=0.393)  Compute df =n-2 (df=27)  obtained value:  critical value: hrt.htm 16

 Step 7 and 8: make decisions  What could be your decision? And why, how to interpret?  obtained value: > critical value: (level of significance: 0.05)  Coefficient of determination is 0.154, indicating that 15.4% of the variance is accounted for and 84.6% of the variance is not.  There is a 5% chance that the two variables are not related at all 17

 Two variables are related to each other One causes another  having a great marriage cannot ensure that the parent-child relationship will be of a high quality as well;  The two variables maybe correlated because they share some traits that might make a person a good husband or wife and also a good parent;  It’s possible that someone can be a good husband or wife but have a terrible relationship with his/her children. 18

 a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be.  These examples indicate that the correlation coefficient, as a summary statistic, cannot replace the individual examination of the data. 19

 To investigate the effect of a new hay fever drug on driving skills, a researcher studies 24 individuals with hay fever: 12 who have been taking the drug and 12 who have not. All participants then entered a simulator and were given a driving test which assigned a score to each driver as summarized in the below figure.  Explain whether this drug has an effect or not? 20