### 统计代写|生物统计学作业代写Biostatistics代考| NORMAL DISTRIBUTION

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|生物统计学作业代写Biostatistics代考|Shape of the Normal Curve

The histogram of Figure $2.3$ is reproduced here as Figure $3.1$ (for numerical details, see Table 2.2). A close examination shows that in general, the relative frequencies (or densities) are greatest in the vicinity of the intervals $20-29,30-$ 39 , and $40-49$ and decrease as we go toward both extremes of the range of measurements.Figure $3.1$ shows a distribution based on a total of 57 children; the frequency distribution consists of intervals with a width of $10 \mathrm{lb}$. Now imagine that we increase the number of children to 50,000 and decrease the width of the intervals to $0.01 \mathrm{lb}$. The histogram would now look more like the one in Figure $3.2$, where the step to go from one rectangular bar to the next is very small. Finally, suppose that we increase the number of children to 10 million and decrease the width of the interval to $0.00001 \mathrm{lb}$. You can now imagine a histogram with bars having practically no widths and thus the steps have all but disappeared. If we continue to increase the size of the data set and decrease the interval width, we eventually arrive at a smooth curve superimposed on the histogram of Figure $3.2$ called a density curve. You may already have heard about the normal distribution; it is described as being a bell-shaped distribution, sort of like a handlebar moustache, similar to Figure 3.2. The name may suggest that most dis-

tributions in nature are normal. Strictly speaking, that is false. Even more strictly speaking, they cannot be exactly normal. Some, such as heights of adults of a particular gender and race, are amazingly close to normal, but never exactly.
The normal distribution is extremely useful in statistics, but for a very different reason – not because it occurs in nature. Mathematicians proved that for samples that are “big enough,” values of their sample means, $\bar{x}^{\prime} s$ (including sample proportions as a special case), are approximately distributed as normal, even if the samples are taken from really strangely shaped distributions. This important result is called the central limit theorem. It is as important to statistics as the understanding of germs is to the understanding of disease. Keep in mind that “normal'” is just a name for this curve; if an attribute is not distributed normally, it does not imply that it is “abnormal.” Many statistics texts provide statistical procedures for finding out whether a distribution is normal, but they are beyond the scope of this book.

From now on, to distinguish samples from populations (a sample is a subgroup of a population), we adopt the set of notations defined in Table 3.7. Quantities in the second column $\left(\mu, \sigma^{2}\right.$, and $\pi$ ) are parameters representing numerical properties of populations; $\mu$ and $\sigma^{2}$ for continuously measured information and $\pi$ for binary information. Quantities in the first column $\left(\bar{x}, s^{2}\right.$, and $\left.p\right)$ are statistics representing summarized information from samples. Parameters are fixed (constants) but unknown, and each statistic can be used as an estimate for the parameter listed in the same row of the foregoing table. For example, $\bar{x}$ is used as an estimate of $\mu$; this topic is discussed in more detail in Chapter 4. A major problem in dealing with statistics such as $\bar{x}$ and $p$ is that if we take a different sample-even using the same sample size-values of a statistic change from sample to sample. The central limit theorem tells us that if sample sizes are fairly large, values of $\bar{x}$ (or $p$ ) in repeated sampling have a very nearly normal distribution. Therefore, to handle variability due to chance, so as to be able to declare-for example-that a certain observed difference is more than would occur by chance but is real, we first have to learn how to calculate probabilities associated with normal curves.

## 统计代写|生物统计学作业代写Biostatistics代考|Areas under the Standard Normal Curve

A variable that has a normal distribution with mean $\mu=0$ and variance $\sigma^{2}=1$ is called the standard normal variate and is commonly designated by the letter $Z$. As with any continuous variable, probability calculations here are always

concerned with finding the probability that the variable assumes any value in an interval between two specific points $a$ and $b$. The probability that a continuous variable assumes a value between two points $a$ and $b$ is the area under the graph of the density curve between $a$ and $b$; the vertical axis of the graph represents the densities as defined in Chapter 2. The total area under any such curve is unity (or $100 \%$ ), and Figure $3.4$ shows the standard normal curve with some important divisions. For example, about $68 \%$ of the area is contained within $\pm 1$ :
$$\operatorname{Pr}(-1<z<1)=0.6826$$
and about $95 \%$ within $\pm 2$ :
$$\operatorname{Pr}(-2<z<2)=0.9545$$
More areas under the standard normal curve have been computed and are available in tables, one of which is our Appendix B. The entries in the table of Appendix B give the area under the standard normal curve between the mean $(z=0)$ and a specified positive value of $z$. Graphically, it is represented by the shaded region in Figure 3.5.

Using the table of Appendix B and the symmetric property of the standard normal curve, we show how some other areas are computed. [With access to some computer packaged program, these can be obtained easily; see Section 3.5. However, we believe that these practices do add to the learning, even though they may no longer be needed.]

## 统计代写|生物统计学作业代写Biostatistics代考|Normal Distribution as a Probability Model

The reason we have been discussing the standard normal distribution so extensively with many examples is that probabilities for all normal distributions are computed using the standard normal distribution. That is, when we have a normal distribution with a given mean $\mu$ and a given standard deviation $\sigma$ (or

variance $\sigma^{2}$ ), we answer probability questions about the distribution by first converting (or standardizing) to the standard normal:
$$z=\frac{x-\mu}{\sigma}$$
Here we interpret the $z$ value (or $z$ score) as the number of standard deviations from the mean.

Example 3.7 If the total cholesterol values for a certain target population are approximately normally distributed with a mean of $200(\mathrm{mg} / 100 \mathrm{~mL})$ and a standard deviation of $20(\mathrm{mg} / 100 \mathrm{~mL}$ ), the probability that a person picked at random from this population will have a cholesterol value greater than $240(\mathrm{mg} / 100 \mathrm{~mL})$ is
\begin{aligned} \operatorname{Pr}(x \geq 240) &=\operatorname{Pr}\left(\frac{x-200}{20} \geq \frac{240-200}{20}\right) \ &=\operatorname{Pr}(z \geq 2.0) \ &=0.5-\operatorname{Pr}(z \leq 2.0) \ &=0.5-0.4772 \ &=0.0228 \text { or } 2.28 \% \end{aligned}
Example 3.8 Figure $3.11$ is a model for hypertension and hypotension (Journal of the American Medical Association, 1964), presented here as a simple illustration on the use of the normal distribution; acceptance of the model itself is not universal.

Data from a population of males were collected by age as shown in Table 3.9. From this table, using Appendix B, systolic blood pressure limits for each group can be calculated (Table $3.10$ ). For example, the highest healthy limit for the $20-24$ age group is obtained as follows.

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## MATLAB代写

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