Essentials

INTRODUCTION TO NULL HYPOTHESIS TESTING 31semble the ND. But it is fortunate that for roughly bell-shaped, unimodal distributions,even if quite skewed, the SDM begins to look surprisingly similar to theND even for modest sample sizes that are no more than 30 or 40.As an example, let us look at annual income for households in the UnitedStates. The population distribution is very (positively) skewed, as you can see inpanel A of Figure 2.1. However, if you take random samples of 40 householdseach, the means of these samples will form a distribution that is much lessskewed, as shown in panel B of Figure 2.1. It may not be unusual to find onehousehold whose annual income is over $100,000, but if you select 40 householdsat random and average their incomes together it is very unlikely that the mean ofthis sample will exceed $100,000. Even if one of the 40 households has a verylarge income it is likely that the other 39 households will be relatively close to thepopulation mean, thus limiting the influence of that one wealthy family on thesample mean. You may notice that the SDM is not only more symmetric thanthe population distribution of individuals, it is also less spread out. That is thenext point to be discussed.A0Annual income per U.S. householdB0Annual income(averaged over 40 households at a time)Figure 2.1 A, Distribution of annual income for the U.S. population; B, samplingdistribution of the mean for N 40