## 统计代写|假设检验代写hypothesis testing代考|Computing the P-value for a Z-test

A Z-test for testing the mean and proportion values for different types of hypotheses using the critical value procedure was studied. The steps for computing the $P$-value for a Z-test including one-sided and two-sided tests will be illustrated step by step, with examples using the general procedure for hypothesis testing.

Example 6.1: Compute the $P$-value to the left of a $\mathbf{Z}$ value: Compute the $P$-value to the left of a negative $Z$ value $(Z=-1.25)$ (left-tailed test). Use a significance level of $0.01(\alpha=0.01)$

Computing the $P$-value for a left-tailed Z-test can be achieved employing the general procedure for testing a hypothesis.
Step 1: Specify the null and alternative hypotheses

The two hypotheses (the null and alternative) for a left-tailed test can be written as presented in Eq. (6.1):
$$H_0: \mu \geq c \text { vs } H_1: \mu<c$$
The hypothesis in Eq. (6.1) represents a one-tailed test because the alternative hypothesis is $H_1: \mu<c$ (left-tailed), where $c$ is a given value.
Step 2: Select the signilicance level $(\alpha)$ for the study
The level of significance is chosen to be $0.01$.
Step 3: Use the sample information to calculate the test statistic value
The test statistic value of $Z$ is given to be $Z=-1.25$, otherwise we have to calculate it using a formula.

Step 4: Calculate the $P$-value and identify the critical and noncritical regions for the study

The $P$-value for the Z-test can be computed easily by employing the standard normal table (Table A in the Appendix) to calculate the probability of $Z(-1.25)$ or less, which represents the required $P$-value. The area to the left of $-1.25$ is $0.1056$ $(1-0.8944=01056)$ which represents the $P$-value. The shaded area in Fig. $6.1$ represents the required $P$-value.

## 统计代写|假设检验代写hypothesis testing代考|Testing one sample mean when the variance is known: P-value

We have studied hypothesis testing for one sample mean when the sample size is large using the critical value (traditional) procedure. The $P$-value procedure for one sample mean using a Z-test will be illustrated employing the same examples presented in Chapter 2, Z-test for one-sample mean, and solved using the critical value procedure. The three situations of hypothesis testing are covered and the $P$-value are calculated.

Example 6.4: The concentration of cadmium of surface water: Example $2.5$ is reproduced “A professor at an environmental section wanted to verify the claim that the mean concentration of cadmium (Cd) of surface water in Juru River is $1.4(\mathrm{mg} / \mathrm{L})$. He selected 35 samples and tested for cadmium concentration. The collected data showed that the mean concentration of cadmium is $1.6$ and the standard deviation of the population is $0.4$. A significance level of $\alpha=0.01$ is chosen to test the claim. Assume that the population is normally distributed.”

The five steps for conducting hypothesis testing employing the $P$-value procedure can be used to test the hypothesis regarding the mean concentration of cadmium of surface water in Juru River. The results of the $P$-value procedure will be compared with the critical value (traditional) procedure.
Step 1: Specify the null and alternative hypotheses
The two hypotheses regarding the mean concentration of cadmium of surface water in Juru River are presented in Eq. (6.4).
$$H_0: \mu=1.4 \text { vs } H_1: \mu \neq 1.4$$
Step 2: Select the significance level ( $\alpha$ ) for the study
The level of significance is chosen to be $0.01$. The $\mathrm{Z}$ critical values for a twotailed test with $\alpha=0.01$ are $\pm 2.58$.

The two procedures will be used to solve this problem; namely the critical value and $P$-value procedures.
Step 3: Use the sample information to calculate the test statistic value
The test statistic value for the Z-test is used to make a decision regarding the mean concentration of cadmium of surface water in Juru River. The test statistic value using the Z-test formula was calculated to be $2.96$.

# 假设检验代考

## 统计代写|假设检验代写hypothesis testing代考|计算z检验的p值

$$H_0: \mu \geq c \text { vs } H_1: \mu<c$$
Eq.(6.1)中的假设代表一个单尾检验，因为备选假设是$H_1: \mu<c$(左尾)，其中$c$是一个给定值。

$Z$的检验统计值被给定为$Z=-1.25$，否则我们必须使用公式计算

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