【学习笔记】数理统计习题十

Q1: True or false, and state why:

1. If the p-value is 0.03, the corresponding test will reject at the significance level 0.02.
   解： 错误，如果$\alpha <p-value$，则接受原假设

2. If a test rejects at significance level 0.06, then the p-value is less than or equal to 0.06.
   解： 正确，对于$p-value$，当$p\le \alpha$时拒绝原假设, 当$p>\alpha$时接受原假设。因此如果试验在显著性水平0.06处被拒绝，则有$p\le 0.06$

3. The p-value of a test is the probability that the null hypothesis is correct.
   解： 错误，$p-value$是由检验统计量的样本观察值得出的原假设可被拒绝的最小显著性水平

Q2: Mutual funds are investment vehicles consisting of a portfolio of various types of investments. If such an investment is to meet annual spending needs, the owner of shares in the fund is interested in the average of the annual returns of the fund. Investors are also concerned with the volatility of the annual returns, measured by the variance or standard deviation. One common method of evaluating a mutual fund is to compare it to a benchmark, the Lipper Average being one of these. This index number is the average of returns from a universe of mutual funds. The Global Rock Fund is a typical mutual fund, with heavy investments in international funds. It claimed to best the Lipper Average in terms of volatility over the period from 1989 through 2007. Its returns are given in the table below.

Year	Investment Return %	Year	Investment Return %
1989	15.32	1999	27.43
1990	1.62	2000	8.57
1991	28.43	2001	1.88
1992	11.91	2002	-7.96
1993	20.71	2003	35.98
1994	-2.15	2004	14.27
1995	23.29	2005	10.33
1996	15.96	2006	15.94
1997	11.12	2007	16.71
1998	0.37

The standard deviation for the Lipper Average is $11.67\%$. Let ${\sigma }^2$ denote the variance of the population represented by the return
percentages shown in the table above. Consider the test

$$H_0:{\sigma }^2\ge (11.67{)}^{2}vs.H_1:{\sigma }^2<(11.67{)}^2.$$

• If the significance level $\alpha =0.05$, what’s your decision?

• Show up the p-value of your test.

解：
假设回报率$X$服从正态分布(原因见上图)，即$X\sim N(\mu ,{\sigma }^2)$，其中$\mu ,{\sigma }^2$未知。因为$\mu$未知，所以我们选择检验统计量$$T(X_{1:n})={\sigma }_0^2$$