## 统计代写|假设检验代写hypothesis testing代考|BSTA511

2022年10月17日

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## 统计代写|假设检验代写hypothesis testing代考|Hypothesis testing for one-sample variance or standard deviation

Hypothesis testing about the variance or standard deviation can be carried out using a new test called a $\chi^2$ test.

Consider a random sample of size $n$ that is selected from a normally distributed population, a claim regarding the variance (standard deviation) value of the variable of interest can be tested employing the $\chi^2$ test for one sample variance (standard deviation) to make a decision regarding the variance (standard deviation) value. The mathematical formula for computing the test statistic value for one sample variance (standard deviation) employing the $\chi^2$ test is presented in Eq. (5.1).
$$\chi^2=\frac{(n-1) s^2}{\sigma_0^2}$$
Follow chi-square distribution with $(n-1)$ degrees of freedom, where $n$ : the sample size,
$s^2$ : the sample variance, and
$\sigma_0^2$ : the population variance.

Example 5.4: The concentration of total suspended solid of surface water: A researcher at an environmental section wishes to verify the claim that the variance of total suspended solids concentration (TSS) of Beris dam surface water is $1.25(\mathrm{mg} / \mathrm{L})$. Twelve samples were selected and the total suspended solids concentration was measured. The collected data showed that the standard deviation of total suspended solids concentration is $1.80$. A significance level of $\alpha=0.01$ is chosen to test the claim. Assume that the population is normally distributed.

The general procedure for conducting hypothesis testing can be used to make the decision regarding the variance of total suspended solids concentration in the surface water of Beris dam.
Step 1: Specify the null and alternative hypotheses
The population variance of total suspended solids concentration $\left(\sigma^2\right)$ is $1.25$; this claim should be under the null hypothesis because the claim represents equality $(=)$. If the variance of total suspended solids concentration of surface water is not equal to $1.25$, then two cases should be considered; in the first case, the variance of total suspended solids concentration is greater than $1.25$, and in the second case, the variance is less than $1.25$. The two cases (greater than and less than) can be represented mathematically as $\neq$. Thus we can write the two hypotheses (null and alternative) as presented in Eq. (5.2).
$$H_0: \sigma^2=1.25 \text { vs } H_1: \sigma^2=1.25$$
We should make a decision regarding the null hypothesis as to whether the variance of total suspended solids concentration exactly equals $1.25$, or the variance of total suspended solids concentration differs (more or less) from $1.25$.

## 统计代写|假设检验代写hypothesis testing代考|What is the observed significance level?

The observed significance level ( $P$-value) is considered to be the second procedure for making a decision regarding the hypothesis under investigation.

Let us give the concept and definition of $P$-value and then illustrate the procedure of using the $P$-value for hypothesis testing employing examples taken from the environmental field.

Suppose the null hypothesis is correct, the probability of obtaining a more extreme value of the test statistic than the one actually observed is called the observed significance level ( $P$-value). Because $P$-value is a probability value, thus its value falls between 0 and 1 .

We can perform hypothesis testing using the $P$-value procedure using similar steps to those used for the critical value (traditional) procedure. The five steps for conducting hypothesis testing employing the $P$-value procedure are presented below.
Step 1: Specify the null and alternative hypotheses
Step 2: Select the significance level $(\alpha)$ for the study
Step 3: Use the sample information to calculate the test statistic value
Step 4: Calculate the $P$-value and identify the critical and noncritical regions for the study
Step 5: Make a decision using $P$-value and interpret the results
A small $P$-value leads to rejecting the null hypothesis while a large $P$-value leads to not rejecting the null hypothesis.

Reject the null hypothesis if the $P-$ value $\leq \alpha$ and fail to reject the null hypothesis if the $P$-value $>\alpha$.

We discuss the general procedure step by step, supported by examples where necessary.

# 假设检验代写

## 统计代写|假设检验代写假设检验代考|单样本方差或标准差的假设检验

$$\chi^2=\frac{(n-1) s^2}{\sigma_0^2}$$

$s^2$:样本方差，
$\sigma_0^2$:总体方差

$$H_0: \sigma^2=1.25 \text { vs } H_1: \sigma^2=1.25$$

## 统计代写|假设检验代写假设检验代考|观察到的显著性水平是什么?

$P$ -value越小，则拒绝原假设，而$P$ -value越大，则不拒绝原假设

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