error happening. The default value of 5% tends to work reasonably well in many applications at balancing both goals. However, other factors also affect the chance of a type 2 error happening for a specific significance level. For example, the chance of a type 2 error tends to decrease the greater the sample size.
1.7 Random Errors and Prediction
So far, we have focused on estimating a univariate population mean, , and quantifying our uncertainty about the estimate via confidence intervals or hypothesis tests. In this section, we consider a different problem, that of “prediction.” In particular, rather than estimating the mean of a population of ‐values based on a sample, , consider predicting an individual ‐value picked at random from the population.
Intuitively, this sounds like a more difficult problem. Imagine that rather than just estimating the mean sale price of single‐family homes in the housing market based on our sample of 30 homes, we have to predict the sale price of an individual single‐family home that has just come onto the market. Presumably, we will be less certain about our prediction than we were about our estimate of the population mean (since it seems likely that we could be further from the truth with our prediction than when we estimated the mean—for example, there is a chance that the new home could be a real bargain or totally overpriced). Statistically speaking, Figure 1.5 illustrates this “extra uncertainty” that arises with prediction—the population distribution of data values, (more relevant to prediction problems), is much more variable than the sampling distribution of sample means, (more relevant to mean estimation problems).
We can tackle prediction problems with a similar process to that of using a confidence interval to tackle estimating a population mean. In particular, we can calculate a prediction interval of the form “point estimate uncertainty” or “(point estimate uncertainty, point estimate uncertainty).” The point estimate is the same one that we used for estimating the population mean, that is, the observed sample mean, . This is because is an unbiased estimate of the population mean, , and we assume that the individual ‐value we are predicting is a member of this population. As discussed in the preceding paragraph, however, the “uncertainty” is larger for prediction intervals than for confidence intervals. To see how much larger, we need to return to the notion of a model that we introduced in Section 1.2.
We can express the model we have been using to estimate the population mean, , as
In other words, each sample ‐value (the index keeps track of the sample observations) can be decomposed into two pieces, a deterministic part that is the same for all values, and a random error part that varies from observation to observation. A convenient choice for the deterministic part is the population mean, , since then the random errors have a (population) mean of zero. Since is the same for all ‐values, the random errors, , have the same standard deviation as the ‐values themselves, that is, . We can use this decomposition to derive the confidence interval and hypothesis test results of Sections 1.5 and 1.6 (although it would take more mathematics than we really need for our purposes in this book). Moreover, we can also use this decomposition to motivate the precise form of the uncertainty needed for prediction intervals (without having to get into too much mathematical detail).
In particular, write the ‐value to be predicted as , and decompose this into two pieces as above:
Then subtract , which represents potential values of repeated sample means, from both sides of this equation:
(1.1)
Thus, in estimating the population mean, the only error we have to worry about is estimation error, whereas in predicting an individual ‐value, we have to worry about both estimation error and random error.
Recall from Section 1.5 that the form of a confidence interval for the population mean is
The term in this formula is an estimate of the standard deviation of the sampling distribution of sample means, , and is called the standard error of estimation. The square of this quantity, , is the estimated variance of the sampling distribution of sample means, . Then, thinking of as some fixed, unknown constant,
error happening. The default value of 5% tends to work reasonably well in many applications at balancing both goals. However, other factors also affect the chance of a type 2 error happening for a specific significance level. For example, the chance of a type 2 error tends to decrease the greater the sample size.
