Statistics 简明教程

Statistics - One Proportion Z Test

检验统计量是一个 z 值（z），由下式定义。 ${z = \frac{(p - P)}{\sigma}}$ 其中 P 是零假设中总体比例的假设值，p 是样本比例，而 ${\sigma}$ 是抽样分布的标准差。

The test statistic is a z-score (z) defined by the following equation. ${z = \frac{(p - P)}{\sigma}}$ where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and ${\sigma}$ is the standard deviation of the sampling distribution.

检验统计量被定义并由下式给出：

Test Statistics is defined and given by the following function:

Formula

其中——

Where −

${z}$ = Test statistics
${n}$ = Sample size
${p_o}$ = Null hypothesized value
${\hat p}$ = Observed proportion

Example

Problem Statement:

一项调查称，十分之九的医生向患有头痛的患者推荐阿司匹林。为了检验这一说法，随机抽取了 100 名医生进行调查。在这 100 名医生中，有 82 人表示他们推荐服用阿司匹林。这一说法准确吗？使用 alpha = 0.05。

A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim, a random sample of 100 doctors is obtained. Of these 100 doctors, 82 indicate that they recommend aspirin. Is this claim accurate? Use alpha = 0.05.

Solution:

定义零假设和备择假设

Define Null and Alternative Hypotheses

此处 Alpha = 0.05。使用 alpha 为 0.05 进行双尾检验，我们期望我们的分布看起来像这样：

Here Alpha = 0.05. Using an alpha of 0.05 with a two-tailed test, we would expect our distribution to look something like this:

$\statistics\images\one proportion$

这里，我们在每个尾部有 0.025。在我们的 z 表中查找 1 - 0.025，我们发现临界值为 1.96。因此，该双尾检验的决策规则为：如果 Z 小于 -1.96 或大于 1.96，则拒绝原假设。计算检验统计：

Here we have 0.025 in each tail. Looking up 1 - 0.025 in our z-table, we find a critical value of 1.96. Thus, our decision rule for this two-tailed test is: If Z is less than -1.96, or greater than 1.96, reject the null hypothesis.Calculate Test Statistic:

z = -2.667，故结果为我们应该拒绝原假设，从而得出结论，9/10 医生向其患者推荐阿司匹林的说法不准确，z = -2.667，p < 0.05。

As z = -2.667 Thus as result we should reject the null hypothesis and as conclusion, The claim that 9 out of 10 doctors recommend aspirin for their patients is not accurate, z = -2.667, p < 0.05.