### 统计代写|linear regression代写线性回归代考|Samples and Populations

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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|linear regression代写线性回归代考|Samples and Populations

We learned earlier that one way to classify statistics is to distinguish between descriptive and inferential methods. At the heart of inferential statistics is a question: how do we know that what we find using a sample reflects what occurs in a population? Can we infer what happens in a population with information from a sample? For instance, suppose we’re interested in determining who is likely to win the next presidential election in the U.S. Assume only two candidates from whom to choose: Warren and Haley. It would be enormously expensive to ask all people who are likely to vote in the next election their choice of president. But we may take a sample of likely voters and ask them for whom they plan to vote. Can we deduce anything about the population of voters based on this sample? The answer is that it depends on a number of factors. Did we collect a good sample? Were the people who responded honest? Do people change their minds as the election approaches? We don’t have space to get into the many issues involved in sampling, so we’ll just assume that our sample is a good representation of the population from which it is drawn.. ${ }^{13}$ Most important for our purposes is this: inferential statistics include a set of methods designed to help researchers answer questions about a population from a sample. Other forms of inferential statistics are not concerned in the same way with a hypothetical population, though. A growing movement is the use of Bayesian inference. We’ll refer to this later, but an adequate description is outside the scope of this book.

An aim of many statistical procedures is to infer something about the population from patterns found in samples. Yet, the cynical-but perhaps most honest-answer is that we never know if what we found says anything accurate about a population. Recall that the definition of statistics provided earlier mentioned uncertainty; statistics is occasionally called the science of uncertainty. The best we can do given a sample is to offer degrees of confidence that our results reflect characteristics of a population. But what do we mean by population? Populations may be divided into target populations and study populations. Target populations are the group about which we wish to learn something. This might include a group in the future (“I wish to know the average weights of future litters sired by my Siberian Husky”) or in the past. Regardless, we typically try to find a population that closely resembles the target population-this is the study population. Many types of populations exist. For instance, we might be interested in the population of seals living on Seal Island off the coast of South Africa; the population of labradoodles in New York City; or the population of voters in Oregon during the 2020 presidential election. Yet some people, when they hear the term population, think it signifies the U.S. population or some other large group. A sample is a set of items chosen from a population. The best known is the simple random sample. Its goal is to select members from the population so that each has an equal chance of being in the sample. Most of the theoretical work on inferential statistics is based on this type of sample. But researchers also use other types, such as clustered samples, stratified samples, and several others.

## 统计代写|linear regression代写线性回归代考|Sampling Error and Standard Errors

Statistical studies are often deemed valuable because they may be used to deduce something about a population from samples, but keep in mind that researchers usually take only a single sample even though they could conceivably draw many.. ${ }^{14}$ Any sample statistic we compute or test we run must thus consider the uncertainty involved in sampling-the sampling error or the “error” due to using only a portion of a population to estimate a parameter

from that population. ${ }^{15}$ The solution to the problem of uncertainty typically involves using standard errors for test statistics, including the mean, the standard deviation, correlations, medians, and, as we shall see, slope coefficients in LRMs. Briefly, a standard error is an estimate of the standard deviationthe variability-of the sampling distribution. The simplest way to understand this is with an example.

Recall that when we compute the variance or the standard deviation, we are concerned with the spread of the distribution of the variable. But imagine drawing many, many samples from a population and computing a mean for each sample. The result is a sample of means from the population $\left(\bar{x}{i} s\right)$ rather than a sample of observations $\left(x{i} s\right)$. We could then compute a mean of these means, or an overall mean, which should reflect pretty accurately-assuming we do a good job of drawing the samples-the actual mean of the population of observations $\left(\frac{\sum \bar{x}{n i}}{n{s}} \cong \mu\right)$. Let’s expand our examination of puppy litters to help us understand this better.
Litter 1: $[40,45,50,55,60,65,70]$
Litter 2: $[40,45,49,56,60,66,75]$
Litter 3: $[39,55,56,58,61,66,69]$
Litter 4: $[42,44,48,55,57,60,66]$
The means for the litters are $55,56,57$, and 53 . Their average-the mean of the means-is $(55+56+57+53) / 4=55.3$. Suppose the samples exhausted the population of puppies. The population mean is thus 55.4. This is close to the mean of the sample means, off by a skosh because of rounding error.

Imagine if we were to take many more samples of puppies. The means from the samples also have a distribution, which is called the sampling distribution of the means. We could plot these means to determine if they follow a normal distribution. In fact, an important theorem from mathematical statistics states that, as more and more samples are drawn, their means follow a normal distribution even if they come from a non-normally distributed variable in the population (see Chapter 4). This allows us to make important inferential claims about LRMs. We shall learn about these claims in later chapters.

## 统计代写|linear regression代写线性回归代考|Significance Tests

Standard errors are utilized in a couple of ways. First, recall from elementary statistics that when we use, say, a $t$-test, we compare the $t$-value to a table of $p$-values. All else being equal, a larger $t$-value equates to a smaller $p$-value. This approach is known as significance testing $g^{16}$ because we wish to determine

if our results are “significantly” different from some other possible result. ${ }^{17}$ Significance testing using standard errors is an inferential approach because it is designed to deduce something about a population based on a sample. But the term significant does not mean important. Rather, it originally meant that the results signified or showed something. ${ }^{18} \mathrm{~A} p$-value is only one piece of evidence that indicates, at best, that a finding is worthy of further consideration; we should not claim that a low $p$-value demonstrates we have found the answer or that it reveals the “truth” about some relationship in a population (recall the section on best statistical practices in Chapter 1). A worthwhile adage to remember is “statistical significance is not the same as practical significance.” We’ll discuss these issues in more detail later in the chapter.

Let’s consider an interpretation of a $p$-value and how it’s used in a significance test rather than derive its computation. Recall that many statistical exercises are designed to compare two hypotheses: the null and the alternative. The null hypothesis usually claims that the result of some observation or an association in the data is due to chance alone, such as sampling error only, whereas the alternative hypothesis is that the result or association is due to some nonrandom mechanism. Imagine, for instance, we measure weights from the litters of two distinct dog breeds: Siberian Husky and German Shepherd. We compute the two means and find that litter 1’s is 5 ounces more than litter 2’s. Assuming we treat the two litters as samples from target populations of Siberian Husky and German Shepherd puppies, we wish to determine whether or not the 5 -ounce difference suggests a difference in the population means. The null and alternative hypotheses are usually represented as:
Null: $\quad H_{0}^{0}:$ Mean weight, litter $1\left(\mu_{1}\right)=$ Mean weight, litter $2\left(\mu_{2}\right)$
Alternative: $H_{a}:$ Mean weight, litter $1\left(\mu_{1}\right) \neq$ Mean weight, litter $2\left(\mu_{2}\right)$
Another way of stating the null hypothesis is that the mean weight of Siberian Husky puppies is actually the same as the mean weight of German Shepherd puppies in the populations of these dog breeds. Because a hypothesis of zero difference is frequently used, though often implicit, some call it the nil hypothesis. Recall that the most common way to compare means from two independent groups is with a $t$-test. We’ll see a detailed example of this test later. For now, suppose the $t$-test provides a $p$-value of $0.04$. One way to interpret this value is with the following garrulous statement:
If the difference in population means is zero $\left(\mu_{1}-\mu_{2}=0\right)$ and we draw many, many samples from the two populations, we expect to find a Do you recognize how a $p$-value is a type of probability based on a frequentist inference approach? Researchers are prone to making statements such as “since the $p$-value is below the conventional threshold of $0.05$, the $t$-test provides evidence with which to reject the null hypothesis” or it “validates the alternative hypothesis. ${ }^{\prime 19}$ But, as outlined later, such statements should be avoided. The $p$-value provides only one piece of evidence-some argue only a sliver-with which to evaluate hypotheses.

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## MATLAB代写

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