Applied Regression Modeling. Iain Pardoe

Applied Regression Modeling - Iain Pardoe


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is constructed differently).

      We saw in Section 1.2 that a normal distribution model fits the home prices example reasonably well. However, we can see from Figure 1.1 that a standard normal distribution is inappropriate here, because a standard normal distribution has a mean of 0 and a standard deviation of 1, whereas our sample data have a mean of 278.6033 and a standard deviation of 53.8656. We therefore need to consider more general normal distributions with a mean that can take any value and a standard deviation that can take any positive value (standard deviations cannot be negative).

      Let images represent the population values (sale prices in our example) and suppose that images is normally distributed with mean (or expected value), images, and standard deviation, images. This textbook uses this notation with familiar Roman letters in place of the traditional Greek letters, images (mu) and images (sigma), which, in the author's experience, are unfamiliar and awkward for many students. We can abbreviate this normal distribution as images, where the first number is the mean and the second number is the square of the standard deviation (also known as the variance). Then the population standardized images‐value,

equation

      has a standard normal distribution with mean 0 and standard deviation 1. In symbols,

equation

      We are now ready to make a probability statement for the home prices example. Suppose that we would consider a home as being too expensive to buy if its sale price is higher than images. What is the probability of finding such an expensive home in our housing market? In other words, if we were to randomly select one home from the population of all homes, what is the probability that it has a sale price higher than images? To answer this question, we need to make a number of assumptions. We have already decided that it is probably safe to assume that the population of sale prices (images) could be normal, but we do not know the mean, images, or the standard deviation, images, of the population of home prices. For now, let us assume that images and images (fairly close to the sample mean of 278.6033 and sample standard deviation of 53.8656). (We will be able to relax these assumptions later in this chapter.) From the theoretical result above, images has a standard normal distribution with mean 0 and standard deviation 1.

      Next, to find the probability that a randomly selected images is greater than 380, we perform some standard algebra on probability statements. In particular, if we write “the probability that images is bigger than images” as “images,” then we can make changes to images (such as adding, subtracting, multiplying, and dividing other quantities) as long as we do the same thing to images. It is perhaps easier to see how this works by example:

equation

      For further practice of this kind of calculation, suppose that we have a budget of images. What is the probability of finding such an affordable home in our housing market? (You should find it is slightly less than a 10% chance; see Problem 1.10.)

      We can also turn these calculations around. For example, which value of images has a probability of 0.025 to the right of it? To answer this, consider the following calculation:

equation

      So, the value 378 has a probability of 0.025 to the right of it. Another way of expressing this is that “the 97.5th percentile of the variable images is images.”


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