Distributions
Sometimes Normal the overall pattern of a large number of observations is so regular that it can be described by a smooth curve
Four examples of histograms which demonstrate a symmetric, bell-shaped pattern.
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Normal distributions can take on many different means and standard deviations. Only the general bell shape must remain the same.
Properties of the Normal Distribution: 1. Symmetric, bell-shaped 2. Mean, and standard deviation, 3. Area under the curve is 1 Because we will mention normal distributions often, a short notation is helpful. We abbreviate the normal distribution with mean and standard deviation as N (, ).
N(0, 1)
N(3, 2)
N(-2, 0.5)
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Q. Why are the normal distributions important in statistics? Here are two reasons (among many others): 1. normal distributions are good descriptions for some distributions of real data (if you don't believe me, take a look at the above figure!). 2. many statistical inference procedures based on normal distributions work well for other roughly symmetric distributions. (this is an advanced point which will be discussed later in the course if time permits). The 68-95-99.7 rule (a.k.a. the "empirical rule") Although there are many normal curves, they all have common properties. Here is one of the most important.
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Example 1.3.1 Bigger animals tend to carry their young longer before birth. The length of horse pregnancies from conception to birth varies according to a roughly normal distribution with mean 336 days and standard deviation 3 days. Use the 68-95-99.7 rule to answer the following questions. 1. Sketch a picture of the distribution of the length of horse pregnancies from conception to birth. Also, using the short notation discussed earlier, to write an abbreviated representation of this distribution.
2. What percent of horse pregnancies fall between 333 to 339 days? What proportion fall between 330 to 342 days?
3. Almost all (99.7%) of horse pregnancies fall in what range of lengths?
4. What percent of horse pregnancies are shorter than 339 days?
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Standardizing observations As the 68-95-99.7 rule suggests, all normal distributions share many properties. In fact, all normal distributions are the same if we measure in units of size about the mean as center. Changing to units is called standardizing.
A z-score tells us how many standard deviations the original observation falls away from the mean, and in which direction. - Observations larger than the mean are positive when standardized, and observations smaller than the mean are negative. z-scores allow us to directly compare observations from two different "groups". Example 1.3.2 Scores on the Wechsler Adult Intelligence Scale for the 20 to 34 age group are approximately normally distributed with mean 110 and standard deviation 25. Scores for the 60 to 64 age group are approximately normally distributed with mean 90 and standard deviation 25. Sarah, who is 30, scores 135 on this test. Sarah's mother, who is 60, also takes the test and scores 120. Who scored higher relative to her age group, Sarah or her mother? Who has the higher absolute level of the variable measured by the test? At what percentile of their age groups are Sarah and her mother? (That is, what percent of the age group has lower scores?)
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The standard normal distribution
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Using the standard normal table Example 1.3.3 Assume that Z follows the standard normal distribution. Consider the following questions: 1. What percent of observations fall below 1? (i.e., is what the relative frequency of the event Z < 1?)
2. What percent of observations fall below 2? (i.e., what is the relative frequency of the event Z < 2?)
3. What percent of observations fall below 1.47? (i.e., what is the relative frequency of the event Z < 1.47?)
Issue: Note that we can not answer the last question by simply using the 6895-99.7 rule. To answer such a question more generally, we need to refer to a standard normal table. This table is given in the front cover of the text book and the appendix (Table A).
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Example 1.3.4 Find the relative frequency of each of the following events in a standard normal distribution. In each case, sketch a standard normal curve with the area representing the relative frequency shaded. (a) Z -2.25
(b) Z -2.25
(c) Z > 1.77
(d) -2.25 < Z < 1.77
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Normal distribution calculations Example 1.3.5 Using the z-score formula given earlier along with our approach to using the standard normal table, we can answer the following types of questions: 1. Suppose X N (3, 2). Find the proportion of the observations that are less than 4.
2. Suppose X N (2, 3). Find the proportion of the observations that are less than 4.
3. Suppose X N (2, 3). Find the proportion of the observations that are greater than 5.
4. Suppose X N (2, 3). Find the proportion of the observations that are between -4 and 8.
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Using the z-score formula given earlier along with our approach to using the standard normal table, we can answer the following types of questions: Example 1.3.6 Too much cholesterol in the blood increases the risk of heart disease. Young women are generally less afflicted with high cholesterol than other groups. The cholesterol levels for women aged 20 to 34 follow an approximately normal distribution with mean 185 milligrams per deciliter (mg/dL) and standard deviation 39 mg/dL. (a) Cholesterol levels above 240 mg/dL demand medical attention. What percent of young women have levels above 240 mg/dL?
(b) Levels above 200 mg/dL are considered borderline high. What percent of young women have blood cholesterol between 200 and 240 ...