## 统计代写|假设检验代写hypothesis testing代考|What is interval estimation

The sample data are usually used to estimate a population parameter either by a single point or an interval. The concept of point estimation and the concept of confidence interval are given first.

Suppose we would like to estimate the mean concentration of the $\mathrm{pH}$ value of surface water (population mean $\mu$ ), in this case we should select a number of sampling points and measure the $\mathrm{pH}$ value of the surface water at each point and then represent the data by a single value, $\bar{X}$ (sample mean), because the sample mean $(\bar{X})$ is the point estimate of the population mean $(\mu)$. This estimation is called point estimation.
How confident are we that the sample mean is close to the population mean? To answer this question lets us think of a range of values that we expect to include the true population mean. Thus, the mean concentration of $\mathrm{pH}$ of surface water could be any value within two values [the lower $(L)$ and upper $(U)$ limits], in this case, it can be said that we are confident that the mean value falls in the interval with a confidence level $(1-\alpha) 100 \%$ :
$$L \leq \mu \leq U$$
This estimation is called the interval estimate or confidence interval. Another important concept that is associated with confidence interval is called the confidence level; the confidence level, $(1-\alpha) 100 \%$, can be defined as the probability of all possible outcomes that the interval estimate will contain the true population parameter. The confidence level is usually selected to be $0.95$ or $0.99$, but may be another value.
We can interpret the confidence level as “we are confident (at confident level) that the population parameter falls in the interval $L \leq \mu \leq U$.”

Other names are given to the confidence level such as the confidence coefficient or degree of confidence.

We can build confidence intervals for population parameters using the general procedure with four steps.
Step 1: Use the sample data to compute the sample statistic
Step 2: Select the significance level $(\alpha)$ for the study and find the critical values
Step 3: Compute the confidence interval
Step 4: Interpret the results

## 统计代写|假设检验代写hypothesis testing代考|When the sample size is large

Consider a large random sample ( $n \geq 30)$ that is selected from a normally distributed population and $Y$ represents a random variable of interest. A confidence interval regarding the population mean of the variable of interest can be built using the sample data. The mathematical formula for computing the confidence interval for one sample mean is presented in Eq. (7.1).
$$\bar{Y} \pm Z_{\frac{\alpha}{2}}\left[\frac{\sigma}{\sqrt{n}}\right]$$
The confidence interval can be written using another form:
$$\bar{Y}-Z_{\frac{a}{2}}\left[\frac{\sigma}{\sqrt{n}}\right] \leq \mu \leq \bar{Y}+Z_{\frac{\alpha}{2}}\left[\frac{\sigma}{\sqrt{n}}\right]$$
where
$\bar{Y}$ represents the sample mean;
$\mu$ represents the population mean;
$\sigma$ represents the population standard deviation;
$Z_{\frac{a}{2}}$ represents the $Z$ critical value;
$n$ represents the sample size;
$\frac{\sigma}{\sqrt{n}}$ represents the standard error; and
$Z_{\frac{\alpha}{2}}\left[\frac{\sigma}{\sqrt{n}}\right]$ represents the margin of error $(E)$.
Note:

• When the sample size is large and the population standard deviation ( $\sigma$ ) is not provided, then we can use the sample standard deviation $(S)$;
• The margin of error $(E)$ can be used to write the confidence interval $\bar{Y}-E \leq \mu \leq \bar{Y}+E$;
• The lower one-tailed confidence interval is given in Eq. (7.2).
$$\bar{Y}-Z_\alpha\left[\frac{\sigma}{\sqrt{n}}\right] \leq \mu$$
or it can be written as the interval $\left[\bar{Y}-Z_\alpha\left[\frac{\sigma}{\sqrt{n}}\right], \infty\right)$

# 假设检验代考

## 统计代写|假设检验代写hypothesis testing代考|What is interval estimation

$$L \leq \mu \leq U$$

## 统计代写|假设检验代写hypothesis testing代考|When the sample size is large

$$\bar{Y} \pm Z_{\frac{a}{2}}\left[\frac{\sigma}{\sqrt{n}}\right]$$

$$\bar{Y}-Z_{\frac{a}{2}}\left[\frac{\sigma}{\sqrt{n}}\right] \leq \mu \leq \bar{Y}+Z_{\frac{a}{2}}\left[\frac{\sigma}{\sqrt{n}}\right]$$

$\bar{Y}$ 代表样本均值;
$\mu$ 代表总体平均值;
$\sigma$ 表示总体标准差;
$Z_{\frac{a}{2}}$ 代表 $Z$ 临界值;
$n$ 代表样本量；
$\frac{\sigma}{\sqrt{n}}$ 代表标准误; 和
$Z_{\frac{a}{2}}\left[\frac{\sigma}{\sqrt{n}}\right]$ 表示误差范围 $(E)$.

• 当样本量很大且总体标准差 $(\sigma)$ 没有提供，那么我们可以使用样本标准差 $(S)$;
• 误差幅度 $(E)$ 可以用来写置信区间 $\bar{Y}-E \leq \mu \leq \bar{Y}+E$;
• 式中给出了较低的单尾置信区间。(7.2)。
$$\bar{Y}-Z_\alpha\left[\frac{\sigma}{\sqrt{n}}\right] \leq \mu$$
或者可以写成区间 $\left[\bar{Y}-Z_\alpha\left[\frac{\sigma}{\sqrt{n}}\right], \infty\right)$

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