Chapter 5: Analysing the Data |

## Constructing a sampling distributionAn example of how a sampling distribution is constructed is shown for a small population of five scores (0, 2, 4, 6, 8).
Repeated sampling with replacement for different sample sizes is shown to produce different sampling distributions. A sampling distribution therefore depends very much on sample size. As an example, with samples of size two, we would first draw a number, say a 6 (the chance of this is 1 in 5 = 0.2 or 20%. We then put the number back and draw another one. Say this is an 8. The mean of our N=2 sample is now (6 + 8)/2 = 7. We would again put the drawn number back into the population. is calculated using N in the denominator rather than N-1 because we have a population, not a sample. Different values for the sample mean which are possible for each sample size along with the associated probability of each mean value occurring are given. Also included is what you would For a sample of size :
Now, we can plot each sampling distribution using the probability or proportion on the Y-axis to get a better feel for what is happening. Note that the Standard error = S.E.M. = You can verify for yourself that this is true using our three example sampling distributions (to within rounding errors, at least). Thus, with larger sample sizes, we will tend, in the long run, to get more accurate estimates. The larger the size of the samples you take each time, the "tighter" the distribution becomes. The sampling distribution becomes less spread out, or more focussed, around the central value as N increases. You should also note that the larger the sample size, the more "normal-looking" the sampling distribution appears (these last two facts are a direct consequence of the Central Limit Theorem). The values for X become less probable the further away from the population value they are. From the sampling distributions given above we can answer a number of questions about sample means. For example, how likely is it that we could select a sample of size N = 4 from this mini-population to have a mean of = 2.0 or less? From the theoretical distribution for N =4, we can simply count up the probabilities of X = 2.0 or less and find that this probability is p = .112. From 0.0016 + 0.0064 + 0.0160 + 0.0320 + 0.0560 = 0.112 (This is a "one-tailed calculation"). Hence our estimate of the expected probability of a sample mean of this size or less occurring is 0.112. From our previous discussion this is not an unlikely event. Sampling distributions can be constructed for any statistic we like. For instance, for each of our small samples in the exercise, we could have simply recorded the largest number only. We would then get a
For example, the single sample t test is a test of the mean of a distribution and it looks like this Note, the t-test is the |

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