Chapter 5 Hypothesis Tests With Means of Samples Part 1: Sept. 10, 2013 The Distribution of Means • In Ch 4, comparison distributions discussed were distributions of individual scores • Now, interested in mean of a group of scores – Comparison distribution of interest will be distribution of means The Distribution of Means • Consists of means of a very large number of samples of the same size – Each sample randomly taken from the same population of individuals – Each point in distribution is a group mean • This will be your comparison distribution when you have N>1. The Distribution of Means – How is it created? Example… • Population N=100, want sample n=5 test scores • 1st random sample = persons 6, 27, 45, 88, 91 (M=78.4) • 2nd random sample = persons 18, 30, 56, 59, 79 (M=82.5) • Plot 78.4, 82.5, etc. • Should look approximately normally distributed The Distribution of Means • Characteristics - assuming a large N – Its mean is the same as the mean of the population of individuals – Its variance is the variance of the population divided by the number of individuals in each of the samples Mean of distribution of means M Variance of distribution of means 2 2 M N The Distribution of Means • Characteristics – Its standard deviation is the square root of its variance 2 2 M M N N SD of distribution of means also known as standard error (σM). ( How much the means of samples are ‘in error’ as estimate of mean of the population.) The Distribution of Means – Shape: it is approximately normal if either • Each sample is of 30 or more individuals or • The distribution of the population of individuals is normal – Example… Hypothesis Testing With a Distribution of Means • Distribution of means will be your comparison distribution • 1) Find a Z score of your sample’s mean on a distribution of means • Z score formula conceptually same as before, but now refers to means of sample & comparison distrib Sample mean (M M ) Z M Mean of distrib of means Std dev of distrib of means Example • Your sample’s mean is 220 (n=64), distribution of means has mean=200, std dev = 6. Example - #9 from practice prob. (p. 182) • 1-sample z test: • Used 1-10 scale to indicate fault of driver in accident. – Population distribution has = 5.5 and = .8 – Here, 16 students rated fault when asked how likely the driver who crashed into other was at fault? – This sample (n=16) had mean=5.9. – Did the manipulation significantly increase fault results? • that is, compare our sample mean to the pop mean • is 5.9 significantly higher than 5.5?