Technical Mathematics, Sixth Edition

21.4. Introduction to Probability

Why do we need probability to learn statistics? In Sec. 21–3 we learned how to compute certain sample statistics, such as the mean and the standard deviation. Knowing, for example, that the mean height of a sample of students is 69.5 in., we might infer that the mean height μ of the entire population is 69.5 in. But how reliable is that number? Is μ equal to 69.5 in. exactly? We'll see later that we give population parameters such as μ as a range of values, say, 69.5 ± 0.6 in., and we state the probability that the true mean lies within that range. We might say, for example, that there is a 68% chance that the true value lies within the stated range.

Further, we may report that the sample standard deviation is, say, 2.6 in. What does that mean? Of 1000 students, for example, how many can be expected to have a height falling within, say, 1 standard deviation of the mean height at that college? We use probability to help us answer questions such as that.

In addition, statistics are often used to "prove" many things, and it takes a knowledge of probability to help decide whether the claims are valid or are merely a result of chance.

Example 29: Suppose that a nurse measures the heights of 14-year-old students in a town located near a chemical plant. Of 100 students measured, she finds 75 students to be shorter than 59 in. Parents then claim that ...