[...]...22 QUANTITATIVE MODELS IN PSYCHOLOGY If you flip the coin 10 times and get nine heads, you know this is a much more likely occurrence if the coin is fixed to give too many heads I use this sort of comparison between sampling distributions in Chapter 3 to explain the concept of power For now I hope you are starting to see how a sampling distribution is useful for understanding the correspondence... months, an intelligence test is 1In the psychological literature, dimensional variables are often referred to using more specific mathematical terms, such as continuous, ordinal, interval, or ratio variables I discuss why this practice is often technically incorrect in Chapter 8 Preliminary Concepts in Inference administered that is believed to have a mean score of 100 and a standard deviation of 15 in the... in statistics to mean “best estimate of”) based on the sample ˆ One formula for σ is N ˆ σ = ∑ (Y i =1 i −Y) 2 , (1.6) N −1 (1.7) so another formula for tY is ¯ tY = Y −μ N ∑ (Yi − Y ) 2 i =1 N ( N − 1) The new denominator of t involves taking each score in the sample, subtracting the sample mean, squaring the difference, summing those 29 30 QUANTITATIVE MODELS IN PSYCHOLOGY squared values, dividing... objects and provides models for failure rates However, it has many other uses F Hypergeometric Poisson Weibull Preliminary Concepts in Inference Conclusion In response to increasing concern about the problem of sampling error, scientists in the 18th century turned to a concept developed by mathematicians interested in probability theory called the sampling distribution The sampling distribution provided... binomial distribution is no longer relevant, and the probability of each value in the population is no longer particularly interesting For example, the probability of each height in the American population is actually less informative than the population mean because the former involves an overwhelming amount of information whereas the latter captures an important aspect of that information in a single... treatment enhances intellectual functioning 2 Identify the null state In the study described, the null, or no-effect, state would occur if the nutritional supplement has no effect on intellectual functioning 3 Convert this null state into a statement about a parameter If the nutritional treatment has no effect on intellectual functioning, then the mean intelligence test score in the population of individuals... of this chapter 35 36 QUANTITATIVE MODELS IN PSYCHOLOGY Fisher’s Model Significance testing is a procedure Fisher introduced for making inferential statements about populations Note that significance testing is not in itself a statistical technique; it is not even a necessary adjunct to the use of statistics It is a logical model that Fisher proposed as a formal approach to addressing questions about populations... deciding whether the nutritional supplement improves intellectual functioning The sample had a mean score of 102.5 If the treatment is ineffective, and µ = 100, then the additional 2.5 points is just sampling error If instead the treatment does improve intellectual functioning, then the additional 2.5 points is due at least in part (because there is still sampling error) to the treatment The question is how... improve quality at the Guinness brewery in Dublin In doing so, he confronted this issue of computing probabilities for means when the population standard deviation is unknown In 1908, he offered a solution based on a new statistic, the t statistic (subsequently modified by Sir Ronald Fisher, an individual who will appear often in this story), and the corresponding sampling distribution, the t distribution... has increased mean intelligence test score For example, in Figure 1.3 I have provided a sampling distribution of the mean that would fit this situation Notice that the y-axis for the binomial distributions in Figures 1.1 and 1.2 reads “Probability,” but in Figure 1.3 it refers to the “Probability Density.” This has to do with a feature of some of the sampling distributions I will discuss The sampling . measurement many times in order to minimize error further. By the late 18th century, astronomers were dealing with situations in which error was a serious problem. Chemistry and physics were experimental. degrees of freedom, the tails of the sampling distribution are pulled in toward the center point, reflecting less variability, a smaller standard error, and less sampling error. To summarize, using. PSYCHOLOGY 1235 9-0 2_PT1-Ch01_rev2.qxd 2/17/11 11:30 AM Page 30 As the degrees of freedom increase, the standard error of the t distribution gets smaller. Standard error is the term used to refer to the standard