When we calculated the probability of getting a single value we used the formula: .
Notice that the distance a value falls from the mean is mediated by how much variability there is overall within the distribution (i.e. the standard deviation).
When we calculate the probability of getting a sample value we will use the formula: .
Notice that the denominator is our estimate of the standard error. That is, the distance a sample mean falls from the mean of the population is mediated by how much variability there is from sample to sample. With z-scores we calculated the probability of observing a single value. With the standard error added into the formula, we are calculating the probability of observing a sample value. Recall from class that if it is relatively unlikely to observe a certain sample (p<.05 for alpha=.05), then we can conclude that the sample did not come from the population.
In the following demonstration we test the hypothesis that a sample of school children sprayed with DDT during the Vietnam War have impaired learning. The mean I.Q. for the general population is 100 with a standard deviation of 20. The sample of children (n=30) has a mean I.Q. score of 90 with a standard deviation of 25. Can we conclude that the sample of children have an I.Q. deficit? Assume alpha = .05