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拼布店

Quilt Shop

专题
Statistics / 统计
难度
L2

题目详情

一家拼布店出售三种不同颜色的拼布：蓝色、红色和绿色。在前 1000 条售出的拼布中，有 400 条是绿色。请计算用于判断顾客是否偏好绿色拼布的合适检验统计量。假设随机抽样、方差齐性，且偏好近似服从正态分布。

A quilt shop sells three different colored quilts: blue, red, and green. Of the first 1000 quilts sold, 400 were green. Calculate the appropriate test statistic to determine if customers have a preference for green quilts. Assume random sampling, variance homogeneity, and that preference is approximately normally distributed.

解析

我们检验的假设为：

H0:p=13,  Ha:p>13H_0 : p = \frac{1}{3}, \ \ H_a : p > \frac{1}{3}

当 n=1000n = 1000 时，样本比例为：

p^=4001000=0.4\hat{p} = \frac{400}{1000} = 0.4

Z 统计量计算为：

Z=p^−pp(1−p)n=0.40−1313(1−13)1000≈4.47Z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} = \frac{0.40 - \frac{1}{3}}{\sqrt{\frac{\frac{1}{3}(1-\frac{1}{3})}{1000}}} \approx 4.47

Original Explanation

We are testing the following hypotheses:

H0:p=13,  Ha:p>13H_0 : p = \frac{1}{3}, \ \ H_a : p > \frac{1}{3}

The sample proportion for n=1000n = 1000 is:

p^=4001000=0.4\hat{p} = \frac{400}{1000} = 0.4

The Z statistic can be calculated as:

Z=p^−pp(1−p)n=0.40−1313(1−13)1000≈4.47Z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} = \frac{0.40 - \frac{1}{3}}{\sqrt{\frac{\frac{1}{3}(1-\frac{1}{3})}{1000}}} \approx 4.47