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# 统计代写|统计推断代考Statistical Inference代写|MS-C1620 Sample size determination

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## 统计代写|统计推断代考Statistical Inference代写|Sample size determination

Setting: In the large-sample scenarios considered in this section, suppose we would test
$$\begin{gathered} H_0: \theta=\theta_0 \ \quad \text { versus } \ H_a: \theta>\theta_0 \end{gathered}$$
using a rejection region of the form
$$\mathrm{RR}={\widehat{\theta}>k},$$
where $k$ is chosen to ensure the test is (approximately) level $\alpha$; i.e., $\alpha \approx P_{H_0}(\widehat{\theta}>k)$. Our goal is to determine the sample size $n$ that confers a Type II Error probability equal to $\beta$ for a pre-specified value $\theta_a>\theta_0$, that is,
$$\theta_a=\theta_0+\Delta,$$
where $\Delta>0$ is the practically important difference we wish to detect. The Type I Error probability request implies
$$\alpha \approx P_{H_0}(\widehat{\theta}>k)=P_{H_0}\left(Z>\frac{k-\theta_0}{\sigma_{\widehat{\theta}}}\right) \Longrightarrow \frac{k-\theta_0}{\sigma_{\widehat{\theta}}}=z_\alpha .$$

## 统计代写|统计推断代考Statistical Inference代写|Hypothesis tests arising from normal populations

Preview: We now derive hypothesis tests for means and variances when the population distribution is $\mathcal{N}\left(\mu, \sigma^2\right)$. We consider one and two populations. Given the duality between hypothesis tests and interval estimators, one should compare this section with Section $8.7$ (notes) in STAT 512. All hypothesis tests in this section are exact; i.e., we are not appealing to large-sample arguments like we did in the last section. Rejection regions come from the $t, \chi^2$, and $F$ distributions, and they have Type I Error probability equal to $\alpha$ exactly.

Remark: Test statistics for all scenarios considered this section are given without proof. This is because we have derived all relevant sampling distributions in STAT 512 (see Chapters 7-8). All tests presented will assume a two-sided alternative (so that the rejection region is two-sided). Rejection regions for one-sided alternatives are formed in the obvious way.
10.4.1 Population mean $\mu$
Setting: Suppose $Y_1, Y_2, \ldots, Y_n$ is an iid sample from a $\mathcal{N}\left(\mu, \sigma^2\right)$ population distribution, where both $\mu$ and $\sigma^2$ are unknown. The goal is to construct a level $\alpha$ test for
$$\begin{gathered} H_0: \mu=\mu_0 \ \quad \text { versus } \ H_a: \mu \neq \mu_0 . \end{gathered}$$

When $H_0$ is true, we know
$$T=\frac{\bar{Y}-\mu_0}{S / \sqrt{n}} \sim t(n-1) .$$
Therefore, a level $\alpha$ test uses the rejection region
$$\mathrm{RR}=\left{t<-t_{n-1, \alpha / 2} \text { or } t>t_{n-1, \alpha / 2}\right}=\left{|t|>t_{n-1, \alpha / 2}\right},$$
where
\begin{aligned} -t_{n-1, \alpha / 2} & =\text { lower } \alpha / 2 \text { quantile of } t(n-1) \ t_{n-1, \alpha / 2} & =\text { upper } \alpha / 2 \text { quantile of } t(n-1) \end{aligned}
see Figure $10.13$ (above). This inference procedure is called a one-sample $t$ test.

# 统计推断代写

## 统计代写|统计推断代考Statistical Inference代写|Sample size determination

$$H_0: \theta=\theta_0 \quad \text { versus } H_a: \theta>\theta_0$$

$$\mathrm{RR}=\hat{\theta}>k,$$

$$\theta_a=\theta_0+\Delta,$$

$$\alpha \approx P_{H_0}(\hat{\theta}>k)=P_{H_0}\left(Z>\frac{k-\theta_0}{\sigma_{\hat{\theta}}}\right) \Longrightarrow \frac{k-\theta_0}{\sigma_{\hat{\theta}}}=z_\alpha .$$

## 统计代写|统计推断代考Statistical Inference代写|Hypothesis tests arising from normal populations

10.4.1 总体平均值 $\mu$

$$H_0: \mu=\mu_0 \quad \text { versus } H_a: \mu \neq \mu_0 \text {. }$$

$$T=\frac{\bar{Y}-\mu_0}{S / \sqrt{n}} \sim t(n-1) .$$

\left 缺少或无法识别的分隔符

$$-t_{n-1, \alpha / 2}=\text { lower } \alpha / 2 \text { quantile of } t(n-1) t_{n-1, \alpha / 2}=\text { upper } \alpha / 2 \text { quantile of } t(n-1)$$

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## MATLAB代写

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