统计题3

可汗统计课45

In a local teaching district a technology grant is available to teachers in order to install a cluster of four computers in their classrooms. From the 6250 teachers in the district, 250 were randomly selected and asked if they felt that computers were an essential teaching tool for their classroom. Of these selected, 142 teachers felt that computers were an essential teaching tool.
1. Caldulate a 99% confidence interval for the proportion of teachers who felt that computers are an essential teaching tool.
2. How could the survey be changed to narrow the confidence interval but to maintain the 99% confidence interval?

分析：

通过样本来估算总体的置信区间。

99%置信区间，意思是给一个同意教师比例的区间，比如50%-60%，认为有99%的概率会有50%-60%的老师同意电脑必要。

用样本来估计总体的理论是中心极限定理，核心是均值相等，方差差一个样本容量的倍数，即均值一样，偏离多远取决于样本容量。

计算：

样本：142/250=56.8% 以同意为1，不同意为0，样本均值56.8%

样本方差作为总体估计时，方差S2=[142(1-0.568)2+108(0-0.568)2]/(250-1)=0.246

样本均值的抽样分布为正态分布，方差σ2≈0.246/250，标准差σ≈0.031

z table 查出99%要求的z=2.58个标准差，即0.08，

总体均值设为μ，即任一样本均值落在[μ-0.08, μ+0.08]的概率为99%，

本次抽样均值为0.568，落在[μ-0.08, μ+0.08]的概率为99%，

所以μ落在[0.488,0.648]的概率为99%。

要想缩小区间，即缩小标准差，根据公式，需要增大样本容量。