# 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|CS5850

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Formulating a Hypothesis

We now return to the problem that an experiment can result only in rejection or nonrejection of a hypothesis. Therefore, we may end up not rejecting an invalid hypothesis. What guidelines should we use in choosing a hypothesis?

The standard technique is to formulate a null hypothesis that we believe is sufficient to explain the data unless statistical evidence strongly indicates otherwise. The null hypothesis should be formulated conservatively, that is, preserving the status quo, where this is applicable. A good way to think about this is in terms of a criminal trial. The judicial system starts with the presumption of innocence. It is up to the prosecution to prove that the defendant is guilty. If the prosecution cannot prove beyond reasonable doubt that the defendant is guilty, the defendant is released. No doubt, this will let some guilty parties go unpunished. But it is preferable to the alternative, whereby the defendant is assumed guilty and must prove innocence.
In formulating a null hypothesis, it is necessary to be precise. In the words of Sir R. A. Fisher, the inventor of this approach, the null hypothesis should be “free from vagueness and ambiguity.” Otherwise, it may be impossible to reject it, making our effort fruitless. Moreover, a hypothesis should be about a population parameter, not a sample, unless the sample includes the entire population.

Consider a router that can execute either scheduling algorithm A or scheduling algorithm B. Suppose that our goal is to show that scheduling algorithm $\mathrm{A}$ is superior to scheduling algorithm $\mathrm{B}$ for some metric. An acceptable conservative null hypothesis would be “Scheduling algorithm $\mathrm{A}$ and scheduling algorithm B have identical performance.” Given this assumption, we would expect the performance metrics for both scheduling algorithms to be roughly the same (i.e., our expectation about the state of the world). If our experiments show this to be the case-for example, if the sample means of the performance metrics for both scheduling algorithms were nearly identical-we would conclude that we do not have sufficient evidence to prove that scheduling algorithm B improved the system, a conservative and scientifically valid decision.

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Comparing an Outcome with a Fixed Quantity

To develop intuition, let us start with a simple example. Suppose that physical considerations lead us to expect that the population mean of a population under study is 0 . Assume that there is no particular status quo that we are trying to maintain. Therefore, a reasonable null hypothesis is
$H_{0}$ : the population mean is 0

To test this hypothesis, we need to sample the population multiple times and compute the corresponding sample means. If the number of samples is large, the central limit theorem implies that each sample mean will be drawn from a normal distribution. We can use this fact to compute its confidence interval-say, at the 95\% level-using the techniques in Section 2.3. We then check whether 0 lies within this interval. One of two cases arises.

1. If 0 lies in the $95 \%(99 \%)$ confidence interval of the sample mean, we cannot reject the null hypothesis. This is usually incorrectly interpreted to mean that with $95 \%(99 \%)$ confidence, the population mean is indeed 0. Of course, all we have shown is that, conditional on the population mean being 0 , the outcome of this experiment has a likelihood greater than $95 \%$ ( $99 \%$ ) (it is consistent with the null hypothesis).
2. If 0 does not lie in the $95 \%(99 \%)$ confidence interval of the sample mean, we reject the null hypothesis. This is usually incorrectly interpreted to mean that, with high confidence, the population mean is not 0 . Again, all we have shown is that, conditional on the mean being 0 , the outcome we saw was rather unlikely, so we have good reason to be suspicious of the null hypothesis.
This example is easily generalized. Suppose that we want to establish that the population mean is $\mu_{0}$. We compute the sample mean $\bar{x}$ as before. Then, we test the hypothesis:
$$H_{0}:\left(\bar{x}-\mu_{0}\right)=0$$

## 电子工程代写|计算数学基础代写Mathematical Foundations of Computing代考|Comparing an Outcome with a Fixed Quantity

0 。假设我们没有试图维持特定的现状。因此，一个合理的零假设是 $H_{0}$ : 总体均值为 0

1. 如果 0 位于 $95 \%(99 \%)$ 样本均值的置信区间，我们不能拒绝原假设。这通常被错淏地解释为意味着 $95 \%(99 \%)$ 置信度，总体均值确实为 0 。当然，我们所展示的是，在总体均值为 0 的条件下，该实验 的结果的可能性大于 $95 \%$ ( $99 \%$ ) (与原假设一致)。
2. 如果 0 不在 $95 \%(99 \%)$ 样本均值的置信区间，我们拒绝原假设。这通常被错误地解释为具有高置信度 的总体均值不是 0 。同样，我们所展示的是，在均值为 0 的条件下，我们看到的结果不太可能，因此 我们有充分的理由怀疑原假设。
这个例子很容易概括。假设我们要确定总体均值是 $\mu_{0}$. 我们计算样本均值 $\bar{x}$ 和以前一样。然后，我们检 验假设：
$$H_{0}:\left(\bar{x}-\mu_{0}\right)=0$$

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