課程名稱︰工程統計學
課程性質︰必修
課程教師︰朱致遠
開課學院：工學院
開課系所︰土木工程學系
考試日期（年月日）︰103/1/7
考試時限（分鐘）：110分鐘
試題 :
Engineering Statistics Final Exam (2014/1/7)
● Exam time: 10:20am-12:10pm.
● Closed-book. Cell phones are strictly prohibited.
● Calculators are allowed.
● The calculation process must be described explicitly, including the
probability functions for the commonly used distributions.
● 30 points total.
1. (4 points) A coin is tossed until a head occurs and the number X of tosses
recorded. After repeating the experiment 256 times, we obtained the
following results:
┌───┬───┬───┬───┬───┬───┬───┬───┬───┐
│x │1 │2 │3 │4 │5 │6 │7 │8 │
├───┼───┼───┼───┼───┼───┼───┼───┼───┤
│f │136 │60 │34 │12 │9 │1 │3 │1 │
└───┴───┴───┴───┴───┴───┴───┴───┴───┘
Conduct the goodness-of-fit test to find out, at the 0.05 level of significance,
whether the observed distribution of X may be fitted by the geometric
distribution g(x; p=1/2), x=1, 2, 3, ….
2. (4 points) Obtain the normal equations (simultaneous equations，聯立方程式)
that can be solved to yield formulas for the least squares estimates β_0,
β_1,β_2, of the regression model, y = β_0 + β_1x + β_2x^2. (You DO NOT
need to solve the equations to yield formulas.)
3. (5 points) A civil engineer investigates the compressive strength of
concrete when mixed with fly ash. The compressive strength for 9 samples in
dry conditions on the twenty-eighth day are as follows (in Mpa):
40.2 30.4 28.9 30.5 22.4 25.8 18.4 14.2 15.3
(a) Find a 99% one-sided lower confidence bound on mean compressive strength.
(b) Find a 98% two-sided confidence interval on mean compressive strength.
(c) Suppose it was discovered that the largest observation 40.2 was
misrecorded and should actually be 20.4. Now the sample mean x=22.9 and the
sample variance s^2=39.83. Use these new values and repeat part (b).
(d) Suppose, instead, it was discovered that the largest observation 40.2 is
correct, but that the observation 25.8 is incorrect and should actually be
24.8. Now the sample mean x=25.0 and the sample variance s^2=70.84. Use
these new values and repeat part (b).
(e) Use the results from (c) and (d) to explain the effect of mistakenly
recorded values on sample estimates. Comment on the effect when the mistaken
values are near the sample mean and when they are not.
4. (4 points) An article describes a study of the thermal inertia properties of
autoclaved aerated concrete used as a building material. Five samples of the
material were tested in a structure, and the average interior temperature
reported was as follows: 23.01, 22.22, 22.04, 22.62, and 22.59.
(a) Test the hypotheses H_0:μ = 22.5 versus H_1:μ≠22.5 with α = 0.05.
Use the P-value approach to answer.
(b) Find a 95% confidence interval on the mean interior temperature and use
it to answer Part (a).
5. (3 points) In semiconductor manufacturing, wet chemical etching is often
used to remove silicon from the backs of wafers prior to metallization. The
etch rate is an important characteristic in this process and known to
follow a normal distribution. Two different etching solutions have been
compared, using two random samples of 10 wafers for etch solution. The
observed etch rates are as follows (in mils/min):
Solution 1: 9.9, 9.4, 9.3, 9.6, 10.2, 10.6, 10.3, 10.0, 10.3, 10.1
Solution 2: 10.2, 10.6, 10.7, 10.4, 10.5, 10.0, 10.2, 10.7, 10.4, 10.3
Find a 95% confidence interval on the difference in mean etch rates.
6. (4 points) The diameter of steel rods manufactured on two different
extrusion machines is being investigated. Two random samples of size n_1=15
and n_2=17 are selected, and the sample means and sample variances are
_ 2 _ 2
x_1 = 8.73, s_1 = 0.035, x_2 = 8.68, and s_2 = 0.40, respectively. Assume
that σ^2 = σ^2 and that the data are drawn from a normal distribution.
(a) Is there evidence to support the claim that the two machines produce
rods with different mean diameters? Use a P-value in arriving at this
conclusion.
(b) Calculate a confidence interval to answer the question in part (a).
7. (2 points) An electrical firm manufactures light bulbs that have a length of
life that is approximately normally distributed with a standard deviation of
40 hours. If a sample of 30 bulbs has an average life of 780 hours, find a
96% confidence interval for the population mean of all bulbs produced by
this firm. How large a sample is needed if we wish to be 96% confident that
our sample mean will be within 10 hours of the true mean?
8. (4 points) Physical fitness testing is an important aspect of athletic
training. A common measure of the magnitude of cardiovascular fitness is the
maximum volume of oxygen uptake during strenuous exercise. A study was
conducted on 24 middle-aged men to determine the influence on oxygen uptake
of the time required to complete a two-mile run. Oxygen uptake was measured
with standard laboratory methods as the subjects performed on a treadmill.
The data are as follows:
┌─────┬────────┬───────┐
│Subject │y, Maximum │ x, Time in │
│ │Volume of O2 │ Seconds │
├─────┼────────┼───────┤
│1 │42.33 │918 │
│2 │53.1 │805 │
│3 │42.08 │892 │
│4 │50.06 │962 │
│5 │42.45 │968 │
│6 │42.46 │907 │
│7 │47.82 │770 │
│8 │49.92 │743 │
│9 │36.23 │1045 │
│10 │49.66 │810 │
│11 │41.49 │927 │
│12 │46.17 │813 │
│13 │46.18 │858 │
│14 │43.21 │860 │
│15 │51.81 │760 │
│16 │53.28 │747 │
│17 │53.29 │743 │
│18 │47.18 │803 │
│19 │56.91 │683 │
│20 │47.8 │844 │
│21 │48.65 │755 │
│22 │53.67 │700 │
│23 │60.62 │748 │
│24 │56.73 │775 │
└─────┴────────┴───────┘
A linear regression model is fitted with Excel and the output is listed below:
摘要輸出
──────
迴歸統計
──────
R 的倍數 0.812338
R 平方 0.659893
調整的 R 平方 0.644433
標準誤 3.49322
觀察值個數 24
────────────
ANOVA
────────────────────────────
自由度 SS MS F 顯著值
────────────────────────────
迴歸 1 520.873 520.873 42.68546 1.43E-06
殘差 22 268.4569 12.20259
總和 23 789.3299
────────────────────────────
────────────────────────────────────────
下限 上限 下限 上限
係數 標準誤 t 統計 P-值 95% 95% 95.0% 95.0%
────────────────────────────────────────
截距 90.8904 6.533019 13.91246 2.21E-12 77.34175 104.4391 77.34175 104.4391
X變數1 -0.05133 0.007857 -6.53341 1.43E-06 -0.06763 -0.03504 -0.06763 -0.03504
────────────────────────────────────────
(a) What is the fitted linear regression model?
(b) What is the coefficient of determination of this model?
(c) Is there a linear relationship between y and x at the 0.05 level of
significance? Why?