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本工具的運算數值是否具備學術與專業參考價值？

是的。所有運算演算法均基於所屬學科公認之國際標準公式編寫，經過嚴格的邊界極限測試，能夠滿足日常工作、課堂學習與工程初步核算之精度要求。

遇到特殊邊界數值或計算結果異常，該如何處理？

請檢查輸入的各項參數單位是否統一、數值是否落於常規物理或邏輯有效區間內（例如數值不能為負數、分母不能為零）。您亦可點選「重設參數」重置為基準值再次測算。

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Z Test For Proportion 計算機｜線上即時試算、公式解析與操作指南

What Is the One-Proportion Z-Test?

The one-proportion z-test (also called a one-sample proportion test) checks whether a population proportion differs from a known or hypothesized value. It is widely used for yes/no, pass/fail, or success/failure data — for example, testing whether a coin is fair, whether a conversion rate beats a benchmark, or whether a defect rate meets a target.

How to Use This Calculator

Enter the number of successes (x), the total sample size (n), and the hypothesized proportion (p₀, a value between 0 and 1). Choose your alternative hypothesis: two-tailed (the true proportion is simply different from p₀), left-tailed (less than p₀), or right-tailed (greater than p₀). The calculator returns the sample proportion, the standard error, the z statistic, and the corresponding p-value.

The Formula Explained

The test statistic is

z = \dfrac{\hat{p} - p_0}{\sqrt{\dfrac{p_0≤ft(1 - p_0\right)}{n}}} \qquad \hat{p} = \dfrac{x}{n}

is the sample proportion. The denominator is the standard error of the proportion computed under the null hypothesis, which is why it uses p₀ rather than p̂. The resulting z is compared to the standard normal distribution to obtain a p-value. The normal approximation is reliable when both n·p_0 and n·(1 - p_0) are at least about 5–10.

Worked Example

Suppose 55 of 100 voters favor a measure and you test p₀ = 0.5 with a two-tailed alternative. Then \hat{p} = 0.55, the standard error

= \sqrt{\dfrac{0.5· 0.5}{100}} = 0.05

and

z = \dfrac{0.55 - 0.5}{0.05} = 1.0

The two-tailed p-value ≈ 0.317, so at \alpha = 0.05 you would not reject the null hypothesis.

使用注意事項與精準校對原則（6 大核心守則）

📐 1. 嚴格對齊輸入量綱與單位

輸入各項數值時請務必確認量綱（例如小時、天數、金額、百分比），避免單位不匹配引發推算偏差。

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📊 3. 符合公認學術與產業計算模型

演算法嚴格遵循國際通用數學公式與標準計算模型，經多重情境交叉核對，數值嚴密精確。

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支援一鍵複製推算結果字串至剪貼簿，亦可直接點擊列印按鈕輸出純淨 PDF 工作報表。

🔄 5. 多情景敏感度動態對比

建議針對極端值與多組預期情景分別試算，觀察輸出數據的波動區間，輔助做出更穩健的決策。

🛡️ 6. 專業諮詢與日常參考免責聲明

本工具運算結果供日常排程、學業研習與事前參考之用，如涉及重大法規或投資決策請諮詢持證專家。

常見問題 (FAQ)

What is a good sample size?

Ensure \(\text{n}\cdot\text{p}_0 \geq 5\) and \(\text{n}\cdot(1 - \text{p}_0) \geq 5\) so the normal approximation holds; otherwise use an exact binomial test.

One-tailed or two-tailed?

Use two-tailed unless you have a directional hypothesis decided before collecting data.

How do I interpret the p-value?

If the p-value is below your significance level (commonly 0.05), reject the null hypothesis that the true proportion equals p₀.

📚 Z Test For Proportion 計算機