2 Chapter 1: Introduction to Statistics

programming skills. The accepted way of avoiding this pitfall is to divide the class members

into the two groups “at random.” This term means that the division is done in such

a manner that all possible choices of the members of a group are equally likely.

At the end of the experiment, the data should be described. For instance, the scores

of the two groups should be presented. In addition, summary measures such as the aver-

age score of members of each of the groups should be presented. This part of statistics,

concerned with the description and summarization of data, is called descriptive statistics.

1.3 INFERENTIAL STATISTICS AND

PROBABILITY MODELS

After the preceding experiment is completed and the data are described and summarized,

we hope to be able to draw a conclusion about which teaching method is superior. This

part of statistics, concerned with the drawing of conclusions, is called inferential statistics.

To be able to draw a conclusion from the data, we must take into account the possibility

of chance. For instance, suppose that the average score of members of the ﬁrst group is

quite a bit higher than that of the second. Can we conclude that this increase is due to the

teaching method used? Or is it possible that the teaching method was not responsible for

the increased scores but rather that the higher scores of the ﬁrst group were just a chance

occurrence? For instance, the fact that a coin comes up heads 7 times in 10 ﬂips does

not necessarily mean that the coin is more likely to come up heads than tails in future

ﬂips. Indeed, it could be a perfectly ordinary coin that, by chance, just happened to land

heads 7 times out of the total of 10 ﬂips. (On the other hand, if the coin had landed

heads 47 times out of 50 ﬂips, then we would be quite certain that it was not an ordinary

coin.)

To be able to draw logical conclusions from data, we usually make some assumptions

about the chances (or probabilities) of obtaining the different data values. The totality of

these assumptions is referred to as a probability model for the data.

Sometimes the nature of the data suggests the form of the probability model that is

assumed. For instance, suppose that an engineer wants to ﬁnd out what proportion of

computer chips, produced by a new method, will be defective. The engineer might select

a group of these chips, with the resulting data being the number of defective chips in this

group. Provided that the chips selected were “randomly” chosen, it is reasonable to suppose

that each one of them is defective with probability p, where p is the unknown proportion

of all the chips produced by the new method that will be defective. The resulting data can

then be used to make inferences about p.

In other situations, the appropriate probability model for a given data set will not be

readily apparent. However, careful description and presentation of the data sometimes

enable us to infer a reasonable model, which we can then try to verify with the use of

additional data.

Because the basis of statistical inference is the formulation of a probability model to

describe the data, an understanding of statistical inference requires some knowledge of