疾病发生率

为了解决这个问题，我们首先要了解我们正在寻找什么以及我们已经获得了哪些信息。

我们将 $A$ 定义为某人患有该疾病的事件，并将 $B$ 定义为该人对该疾病检测呈阳性的事件。我们的目标是找到：

$P(\textrm{Person has the disease} \mid \textrm{Person tests positive}) = P(A \mid B)$

我们已经知道以下内容：

根据发病率： -$P(\textrm{Having the disease}) = P(A) = 0.02$ -$P(\textrm{Not having the disease}) = \neg P(A) = 1 - 0.02 = 0.98$

基于假阴性率：

• $P(\textrm{Testing negative} \mid \textrm{Have the disease}) = P(\neg B \mid A ) = 0.1$

基于误报率： -$P(\textrm{Testing positive} \mid \textrm{Not having the disease}) = P(B \mid \neg A) = 0.01$

鉴于我们已经有了这些先验知识，我们可以应用贝叶斯规则来找到这个问题的解决方案。贝叶斯规则定义为：

$P(A \mid B) = \frac{P(B \mid A) \times P(A)}{P(B)}$

代入我们所知道的结果：

$P(A \mid B) = \frac{P(B \mid A) \times P(A)}{P(B)} = \frac{(1 - P(\neg B|A)) \times P(A)}{(1 - P(\neg B|A)) \times P(A) + P(B \mid \neg A) \times P(\neg A))} = \frac{(1 - 0.1) \times 0.02}{(1 - 0.1) \times 0.02 + (0.01) \times 0.98}$ $P(A \mid B) \approx 0.647$

---

Original Explanation

To solve this question, let's first understand what we are looking to find and what information we are already given.

We'll define $A$ to be the event that a person has the disease and we'll define $B$ to be the event that the person tests positive for the disease. Our goal is to find:

$P(\textrm{Person has the disease} \mid \textrm{Person tests positive}) = P(A \mid B)$

We already know the following:

Based on the incidence rate:

• $P(\textrm{Having the disease}) = P(A) = 0.02$
• $P(\textrm{Not having the disease}) = \neg P(A) = 1 - 0.02 = 0.98$

Based on the false negative rate:

• $P(\textrm{Testing negative} \mid \textrm{Have the disease}) = P(\neg B \mid A ) = 0.1$

Based on the false positive rate:

• $P(\textrm{Testing positive} \mid \textrm{Not having the disease}) = P(B \mid \neg A) = 0.01$

Given that we already have this prior knowledge, we can apply Bayes rule to find the solution to this problem. Bayes rule is defined as:

$P(A \mid B) = \frac{P(B \mid A) \times P(A)}{P(B)}$

Plugging in what we know we get:

$P(A \mid B) = \frac{P(B \mid A) \times P(A)}{P(B)} = \frac{(1 - P(\neg B|A)) \times P(A)}{(1 - P(\neg B|A)) \times P(A) + P(B \mid \neg A) \times P(\neg A))} = \frac{(1 - 0.1) \times 0.02}{(1 - 0.1) \times 0.02 + (0.01) \times 0.98}$ $P(A \mid B) \approx 0.647$

回想一下，显着性水平是指当原假设实际上为真时我们拒绝原假设的概率。因此，我们可以将 $\alpha$ 求解为：

$\alpha = P( X \leq 12 \mid p = 0.8)$

我们首先考虑治愈率为 80% 且恰好 $X = 12$ 人被治愈的概率。这可以表述为

$P(X = 12 \mid p=0.8) = {20 \choose 12} \ 0.8^{12} \times 0.2^{8}$

现在我们可以将之前的发现推广到 $X \leq 12$

$\sum_{i=0}^{12} {20\choose i} \ 0.8^i * 0.2^{20-i}$

---

Original Explanation

Recall, that the significance level refers to the probability that we reject the null hypothesis when it is in fact true. Therefore, we can solve for $\alpha$ as:

$\alpha = P( X \leq 12 \mid p = 0.8)$

Let's first consider the probability that there is an 80% cure rate and exactly $X = 12$ people were cured. This can be formulated as

$P(X = 12 \mid p=0.8) = {20 \choose 12} \ 0.8^{12} \times 0.2^{8}$

Now we can generalize this previous finding for $X \leq 12$

$\sum_{i=0}^{12} {20\choose i} \ 0.8^i * 0.2^{20-i}$

功效为 $1-\beta$，其中 $\beta$ 是当 $H_a$ 为真时检验统计量不在拒绝区域内的概率。为了计算功率，我们必须求解 $\beta$。

$\beta = P(\textrm{We fail to reject } H_0 \mid H_a \textrm{ is true}) = P(x > 12 \mid p=0.6)$

$=\sum_{i=13} ^{20} {20 \choose i} \times 0.6^{i} \times 0.4^{20-i} \approx 0.416$。

因此，本次测试的功效为：$1-\beta \approx 0.584$

---

Original Explanation

Power is $1-\beta$, where $\beta$ is the probability that the test statistic is not in the rejection region when $H_a$ is true. In order to calculate power, we must solve for $\beta$.

$\beta = P(\textrm{We fail to reject } H_0 \mid H_a \textrm{ is true}) = P(x > 12 \mid p=0.6)$

$=\sum_{i=13} ^{20} {20 \choose i} \times 0.6^{i} \times 0.4^{20-i} \approx 0.416$.

Thus, the power of this test is: $1-\beta \approx 0.584$