Chapter 5 Hypothesis Tests With Means of Samples Part 1

The Distribution of Means In Ch 4, comparison distributions discussed were distributions of individual scores –Each point was an individual score 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 1 st random sample = persons 6, 27, 45, 88, 91 (M=78.4) 2 nd 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 (so will be less than variance of population) Mean of distribution of means Variance of distribution of means

The Distribution of Means Characteristics –Its standard deviation is the square root of its variance SD of distribution of means also known as standard error (σ M ). (= 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 –Ex) Find the characteristics of this distribution of means: Population of students taking GRE is normal with μ=500, σ= 100 (note – variance = 10,000). What will be distrib of means for samples=50?

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 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. Z score for this sample’s mean ?

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? (that’s the manipulation…”crashed”) –Our sample (n=16) had mean=5.9. –Did the word ‘crashed’ increase fault results? (that is, compare our sample mean to the pop mean - is 5.9 significantly higher than 5.5?) –We’ll work through this example in class…(or try it before class for a challenge!)