统计题4

可汗统计课47

A neurologist is testing the effect of a drug on response time by injecting 100 rats with a unit dose of the drug, subjecting each to neurological stimulus, and recording its response time. The neurologist knows that the mean response time for rats not injected with the drug is 1.2 seconds. The mean of the 100 injected rats' response time is 1.05 seconds with a sample standard deviation of 0.5 seconds. Do you think that the drug has an effect on response time?

NULL HYPOTHESIS零假设H₀：drug has no effect，即注射药物后的平均反映时间与未注射时相同，仍然是1.2，计算在这种前提下样本为1.05的概率。

样本方差S=0.5

样本均值的抽样分布为正态，假设μ=1.2，σ≈S/√100=0.05

样本值1.05距离均值：|1.05-μ|/σ=3个标准差的距离

因为3个标准差已经是99.7%的概率了，所以样本值比1.05更加极端的概率是0.3%，

即零假设的概率为0.3%，很小，不接受该假设，因此认为drug has effect.

因为仍然有0.3%的概率是药物无效但是样本为1.05，如果确实如此，这叫第一型错误。