Using the confidence interval to test a hypothesis

In the previous chapter the confidence interval was introduced as a device for estimating a population parameter. The interval can also be used to assess the plausibility of a hypothesised value for the parameter. Miller (1951) cites a study of the vocabulary of children in which the average number of words recognised by children aged 6–7 years in the USA was 24,000. Suppose that the same test had been carried out in the same year on 140 British children of the same age and that the mean size of vocabulary recognised by that sample was 24,800 with a sample standard deviation of 4,200 words. How plausible is the hypothesis that the population from which the sample of British children was chosen had the same mean vocabulary as the American children of the same age? Admittedly the sample of British children had a higher mean vocabulary size, but many samples of American children would also have had a mean score of more than 24,000. We need to rephrase the question. The mean of a sample of British children is 24,800, not 24,000. Is it nevertheless plausible that the mean vocabulary of the British population of children in this age range could be 24,000 words and that the apparent discrepancy is simply due to sampling variation, so that a new sample will have a mean vocabulary size closer to, perhaps less than, 24,000?