One Sample Z-test Convert raw scores to z-scores to test hypotheses about sample Using z-scores allows us to match z with a probability Calculate: z = (statistic – parameter)/standard error of statistic = (sample mean – population mean)/standard error of mean where standard error of mean =  /sqrt(n)

One Sample Z-test If the calculated Z falls outside of the “likely” zone of our distribution, we reject the null hypothesis. The area in the distribution that falls outside of the “likely” zone is called the “region of rejection” and is formally called  --- or the probability of making a Type 1 error Typically  is set at.05

 

Types of alternative hypotheses Non-directional – “there is a difference” –Also called “two-tailed” Directional – “there is a difference that is directional” –Also called “one-tailed” Students in CPS have a different average GRE score than OSU graduate students in general Students in CPS have a higher average GRE score than OSU graduate students in general

Types of alternative hypotheses Students in CPS have a different average GRE score than OSU graduate students in general H 0:  =average OSU GRE H 1 :  average OSU GRE Students in CPS have a higher average GRE score than OSU graduate students in general H 0:  =average OSU GRE H 1 :  average OSU GRE

Two tailed hypothesis If we take a sample of 30 students and find their average IQ to be , we can test whether the population from which our sample came had an average IQ of 100 H 0:  =100 (average IQ) H 1 :  ≠ 100 Convert our sample mean to Z statistic ( )/2.74 = 2.4

Z=-2.4 Z= 2.4

One tailed hypothesis If we take a sample of 30 students and find their average IQ to be , we can test whether the population from which our sample came had an average IQ greater than 100 H 0:  =100 (average IQ) H 1 :  > 100 Convert our sample mean to Z statistic ( )/2.74 = 2.4

Z=2.4

One tailed hypothesis If we take a sample of 30 students and find their average IQ to be 93.42, we can test whether the population from which our sample came had an average IQ less than 100 H 0:  =100 (average IQ) H 1 :  < 100 Convert our sample mean to Z statistic ( )/2.74 = - 2.4

Z=-2.4

Steps of Hypothesis Testing 1.State Hypothesis 2.Set Criterion for Rejection H 0 3.Compute Test Statistic and Probability 4.Make Decision

One-Sample T-test If we do not know the  of the population – we can’t calculate the SE of the mean We can estimate it with the SD of the sample: SEmean=s/sqrt(n) Instead of calculating z statistic – we calculate the t-statistic t = (statistic – parameter)/standard error of statistic = (sample mean – population mean)/standard error of mean where standard error of mean = s /sqrt(n)

One-Sample T-test After calculating t-statistic, we find associated p-value SPSS does this for us T is about the same as Z in large samples. For this class we will always use T

Making a decision We compare the calculated p-value to our preset  level If p<  reject null hypothesis If p>  fail to  reject null hypothesis

1-tailed t-test

1-sample t-test As sample size increases, the results from a t-test approximate a z-test SPSS only does one sample t-tests