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1.2: Overview

Skills to Develop

  • To obtain an overview of the material in the text.

The example we gave in the first section seems fairly simple, but it illustrates some significant problems. We supposed that the \(200\) cars of the sample had an average value of \($8,357\) (a number that is precisely known), and concluded that the population has an average of about the same amount, although its precise value is still unknown. What would happen if someone else were to take another sample of exactly the same size from exactly the same population? Would he or she get the same sample average as we did, \($8,357\)? Almost surely not. In fact, if the investigator who took the second sample reported precisely the same value, we would immediately become suspicious of his result. The sample average is an example of what is called a random variable: a number that varies from trial to trial of an experiment (in this case, from sample to sample), and does so in a way that cannot be predicted precisely. Random variables will be a central object of study for us, beginning in Chapter 4.

Another issue that arises is that different samples have different levels of reliability. We have supposed that our sample of size \(200\) had an average of \($8,357\). If a sample of size \(1,000\) yielded an average value of \($7,832\), then we would naturally regard this latter number as probably a better estimate of the average value of all cars, since it came from a larger sample. How can this be expressed? An important idea developed in Chapter 7 is the confidence interval: from the data we will construct an interval of values using a process that has a certain chance, say a \(95\% \) chance, of generating an interval that contains the true population average. Thus, instead of reporting a single estimate, \($8,357\), for the population mean we might say that, based on our sample data, we are \(95\% \) certain that the true average is within \($100\) of our sample mean, that is, we are \(95\% \) certain that the true average is the between \($8,257\) and \($8,457\). The number \($100\) will be computed from the sample data just as the sample mean \($8,357\) was. This "\(95\% \) confidence interval" will automatically indicate the reliability of the estimate that we obtained from the sample. Moreover, to obtain the same chance of containing the unknown parameter, a large sample will typically produce a shorter interval than a small sample will. Thus large samples usually give more accurate results.  Unless we perform a census, which is a "sample" that includes the entire population, we can never be completely sure of the exact average value of the population. The best that we can do is to make statements of probability, an important concept that we will begin to study formally in Chapter 3.

Sampling may be done not only to estimate a population parameter, but to test a claim that is made about that parameter. Suppose a food package asserts that the amount of sugar in one serving of the product is \(14\) grams. A consumer group might suspect that it actually contains more. How would they test the competing claims about the amount of sugar, "\(14\) grams" versus "more than \(14\) grams"? They might take a random sample of perhaps \(20\) food packages, measure the amount of sugar in one serving of each one, and average those amounts. They are not interested in measuring the average amount of sugar in a serving for its own sake; their interest is simply whether the claim about the true amount is accurate. Stated another way, they are sampling not in order to estimate the average amount of sugar in one serving, but to see whether that amount, whatever it may be, is larger than \(14\) grams. Again because one can have certain knowledge only by taking a census, ideas of probability enter into the analysis. We will examine tests of hypotheses beginning in Chapter 8 .

Several times in this introduction we have used the term “random sample.” Generally the value of our data is only as good as the sample that produced it. For example, suppose we wish to estimate the proportion of all students at a large university who are females, which we denote by \(p\). If we select \(50\) students at random and \(27\) of them are female, then a natural estimate is \( p \approx \hat{p} = 27/50 = 0.54 \) or \(54\% \). How much confidence we can place in this estimate depends not only on the size of the sample, but on its quality, whether or not it is truly random, or at least truly representative of the whole population. If all \(50\) students in our sample were drawn from a College of Nursing, then the proportion of female students in the sample is likely higher than that of the entire campus. If all \(50\) students were selected from a College of Engineering Sciences, then the proportion of students in the entire student body who are females could be underestimated. In either case, the estimate would be distorted or biased. In statistical practice an unbiased sampling scheme is important but in most cases not easy to produce. For this introductory course we will assume that all samples are either random or at least representative.

Key Takeaway

  • Statistics computed from samples vary randomly from sample to sample. Conclusions made about population parameters are statements of probability.

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