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一凡007 · 2024年07月20日

什么时候按照单尾算?

NO.PZ2023091601000053

问题如下:

Bob tests the null hypothesis that the population mean is less than or equal to 45. From a population size of 3,000,000 people, 81 observations are randomly sampled. The corresponding sample mean is 46.3 and sample standard deviation is 4.5. What is the value of the appropriate test statistic for the test of the population mean, and what is the correct decision at the 1 percent significance level?

选项:

A.

z = 0.29, and fail to reject the null hypothesis.

B.

z = 2.60, and reject the null hypothesis.

C.

t = 0.29, and accept the null hypothesis

D.

t = 2.60, and neither reject nor fail to reject the null hypothesis.

解释:

A is incorrect. The denominator of the z-test statistic is standard error instead of standard deviation. If the denominator takes the value of standard deviation 4.5, instead of standard error 4.5/sqrt(81), the z-test statistic computed will be z = 0.29, which is incorrect.

B is correct. The population variance is unknown but the sample size is large (>30). The test statistics is: z = (46.3-45)/(4.5/ (sqrt (81)) = 2.60. Decision rule: reject H0 if z(computed)>z(critical). Therefore, reject the null hypothesis because the computed test statistics of 2.60 exceeds the critical z-value of 2.33.

C is incorrect because z-test (instead of t-test) should be used for sample size (81) ≥30.

D is incorrect because z-test (instead of t-test) should be used for sample size (81)≤30.

什么时候按照双尾计算?麻烦老师解答一下

1 个答案
已采纳答案

李坏_品职助教 · 2024年07月20日

嗨,从没放弃的小努力你好:


如果是假设检验(Hypothesis Test):

  1. H0是μ = μ0,就是等于号的这种原假设,这个要用双尾值。因为原假设是等于号,说明拒绝域是左右两侧。
  2. H0是μ ≥或者μ≤,这种带着不等号的原假设,拒绝域只包含左侧或右侧一边,要用单尾值


如果是区间估计(计算confidence interval):区间估计都是在均值±Z倍的标准误,既包括左侧也包括右侧,所以就用双尾值


如果是计算VaR(Value at Risk),这个只考虑左尾的损失值,所以用单尾值

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