東京工業大学 環境・社会理工学院 融合理工学系 2018年8月実施 概率统计

标签：

• TITech
• Independence-of-Random-Variables
• Normal-Distribution
• Binomial-Distribution
• Hypothesis-Testing

Author

思齐塾, 祭音Myyura

Description

The useful life per kilometer of a paved road as the usable time to require repair is approximately described as a normal distribution with a mean of 2.8 years and COV (Coefficient of Variation) of 40%. Assume that the lives among any different one kilometers are statistically independent. Answer the following questions. You can refer to Table 3-1 if needed.

1. What is the probability that one kilometer of paved road will require repair in one year?

2. What is the probability that there will be no repairs required in the first year of a 3-kilometer stretch of paved road?

3. What is the probability that 2 of the 3-kilometer stretch will need repairs in the first year?

4. If 10% is the probability that one kilometer of paved road will require repair, what is the useful life of the road?

5. New pavement materials are introduced in a 25 kilometer section of the paved road as a test. The result of useful life on the road is a mean of 3.2 years and a standard deviation of 1.2 years as a normal distribution. Define the null hypothesis and the alternative hypothesis whether new pavement materials can change the useful life, and perform a two-sided hypothesis test at the 5% significance level.

题目描述

把铺装道路每一公里从投入使用到需要维修的时间称为该公里路段的使用寿命。其分布可近似为均值 $2.8$ 年、变异系数（COV）$40\%$ 的正态分布；任意两个不同的一公里路段的寿命在统计上相互独立。必要时可使用题面所指的表 3-1。回答下列问题。

1. 求一公里铺装道路在一年内需要维修的概率。

2. 对一段三公里长的铺装道路，求第一年内三个一公里路段均不需要维修的概率。

3. 求上述三个一公里路段中恰有两个在第一年内需要维修的概率。

4. 求使一公里路段在该期限内需要维修的概率为 $10\%$ 时所对应的使用寿命，即该寿命分布的第 $10$ 百分位数。

5. 在一段 $25$ 公里铺装道路上试用新铺装材料。试验所得寿命可视为正态分布，样本均值为 $3.2$ 年、标准差为 $1.2$ 年。为判断新材料能否改变使用寿命，写出原假设与备择假设，并在 $5\%$ 显著性水平下进行双侧检验。

Editorial note: The Japanese version of the same examination (tse_201808_tiku_1160.md) gives the standard deviation as $1.2$ years. The former value $12$ in this file was a transcription error caused by a missing decimal point and has been corrected against that counterpart.

Kai

道路1 kmの寿命を $T$ とする。変動係数が40%なので、標準偏差は

$$\sigma=0.40\times2.8=1.12\ \text{年}$$

であり、 $T\sim N(2.8,1.12^2)$ とする。以下、 $\Phi$ は標準正規分布の累積分布関数である。

1.

1年以内に補修を要する確率は

$$P(T\le1)=\Phi\left(\frac{1-2.8}{1.12}\right) =\Phi(-1.6071\ldots)=\boxed{0.0540\ \text{(約)}}.$$

2.

1 kmが1年以内に補修を要しない確率は $1-p$ （ただし $p=0.0540115\ldots$ ）である。3区間は独立なので、どの区間も補修を要しない確率は

$$(1-p)^3=(0.945988\ldots)^3=\boxed{0.8466\ \text{(約)}}.$$

3.

1年以内に補修を要する区間数は二項分布 $\operatorname{Bin}(3,p)$ に従う。したがって、ちょうど2区間が補修を要する確率は

$$\binom32p^2(1-p)=3(0.0540115\ldots)^2(0.945988\ldots) =\boxed{0.0083\ \text{(約)}}.$$

4.

$P(T\le t)=0.10$ となる $t$ を求める。標準正規分布の10%点は $z_{0.10}=-1.28155\ldots$ なので、

$$\frac{t-2.8}{1.12}=-1.28155\ldots$$

より

$$\boxed{t=2.8-1.28155\times1.12=1.365\ldots\ \text{年}\simeq1.36\ \text{年}}$$

である。

5.

Let $\mu$ denote the population mean useful life with the new material. Since the question asks whether the useful life changes, use the two-sided hypotheses

$$\boxed{H_0:\mu=2.8\ \text{ years},\qquad H_1:\mu\ne2.8\ \text{ years}}.$$

Treat the reported standard deviation $1.2$ years as the sample standard deviation $s=1.2$ for the 25 sections. For a normal population with unknown variance, the test statistic is

$$t=\frac{\bar T-2.8}{s/\sqrt n} =\frac{3.2-2.8}{1.2/\sqrt{25}} =\frac53=1.6667$$

which has a $t$ distribution with 24 degrees of freedom under $H_0$. The critical value for a two-sided 5% test is $t_{0.975,24}\simeq2.064$, and

$$|t|=1.6667<2.064.$$

Therefore,

$$\boxed{\text{Do not reject }H_0.\text{ There is no significant evidence that the new material changes the mean useful life.}}$$

The two-sided $p$-value is approximately $0.109$. Thus the result is not significant at the 5% level; this does not prove that the new material has no effect, but only that this sample does not provide sufficient evidence of a change in the mean useful life.