JIANGSU UNIVERSITY OF SCIENCE AND TECHNOLOGY
Course Final Examination
Course Name：Probability & Statistics
2st Semester of 2021-2022
PART I、Single Choice:(3 points per question, 15 points)
Do not answer in the binding line
Name：
1、Which of the following equation is not true?
A. ( A B ) ( A C ) A ( B C )
B. A B
（A B） B
C. A B A B
D. ( A B ) C A B C
2、Suppose X N (0,1) , Y N (1,1) , if X and Y are independent, then
A. P ( X Y 0) 1/ 2
B. P ( X Y 1) 1/ 2
C. P ( X -Y 0) 1/ 2
)
(
)
D. P ( X -Y 1) 1/ 2
3、Assume that population X N (2,16) , and X1,X2 ,,Xn is a sample from X, Then which one is true?(
A. X-2 N (0,1)
B. X-2 N (0,1)
4
C. X-2 N (0,1)
n
6
4、Assume that X1 , X2 ,, Xn is a sample from the population X N (0,1) , X and S
A. X N (0,1)
B.
n X N (0,1)
C.
n
i1
5、To give an interval estimation for the mean
2
are the sample mean and
(
X
2
i
2 (2n )
)
D. X-2 N (0,1)
2 n
sample variance respectively, which one is true?
)
D. X t (n 1)
S
of a population X N ( , 2 ) We get a confidence interval
with confidence level 95%. It means this interval
Student ID：
(
(
)
A. on average contains 95% of the population values. B. on average contains 95% of the sample values.
C. has 95% of the chance to contain the value of .
D. has 95% of the chance to contain the sample values
PART II、Fill in The Blanks:(3 points per question, 15 points)
1、Put 3 balls into 4 boxes randomly. The probability of that there is at most one ball in each box is ___________
Do not answer in the binding line
Major：
2、Suppose Y=2X+5, X N ( , 2 ) . Determine the distribution of Y____________________________
3 、 Assume that X , S 2 are the sample mean and sample variance of the sample of size n respectively from the
population X N ( , 2 ) , then
4、If and 2 are mean and variance of the distribution of a population X, and X , S 2 are the mean and Variance
of a random sample with size n from the population X, then EX =__________and ES 2 =__________
5 、 Assume that the random variables X and Y are mutually independent, and both of them have normal
Distribution N (0,32 ) X1,X2 ,,X9 and Y1 , Y2 , , Y9 represent simple random samples form X and Y ， then the
statistic U
Grade：
2
Xi X
__________________
i 1
n
X 1 X 2 X 9
Y12 Y22 Y92
___________
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PART III Calculation and Proof: (8 points per question,56 points)
1、Four balls are selected at random without replacement from a box containing 3 white balls and 5 blue balls. Find the probability
of the given events.1) Two of the balls are white and two are blue. 2) Exactly three of the balls are blue.
2、A company X has four factories F1 , F2 , F3 , F4 , all of which produce similar automotive parts. Suppose that F1 , F2 , F3 , F4
produce 40%, 30%, 20%, 10% of the total production, respectively. On average, 10%of the parts produce by F1 are not up to
standard. For F2 , F3 , F4 the corresponding percent are 3%, 2% and 5%, respectively. 1)Find the probability that an automotive
part produced by company X is notup to the standard.2)If an automotive part was chosen randomly and found to be defective,
what is the probability that it was made by factory F1 ?
3、If two events A and B are independent, prove that the event A and B are also independent.
2x 0 x
,
, where is unknown parameter. Let
其他
0,
4、Suppose a population has the density function f ( x ) 2
X 1 , X 2 , X n be a random sample of size n taken from population, find the maximum likelihood estimator for .
5 、 In 16 test runs, the gasoline consumption of a normal experimental engine had a standard derivation of 2.2 gallons. Find a
confidence interval for 2 with confidence level 0.99 which measure the true variability of the gasoline consumption of the
engine.
6 、 A manufacture of car batteries claims that a life of his batteries is approximately normally distributed with a standard
deviation equal to 0.9 years. Should we believe that 0.9 years at the 5% level of significance?
7、Suppose T ~ t n ，Proof： T 2 ~ F 1，n
PART IV Analysis description: (14points)
A compony claim that the average life of a certain type of battery is 21.5 hours. A laboratory tests battery manufactured by this
compony and obtains the following latterly lifetime in hours:16,25,19,18,22,20.Do these data indicate that this type of battery has
a shorter average life than that it is claimed by the company (5% level).Please explain each of the following concept according to
this example:
1)null hypothesis; 2)test statistic; 3)significant level; 4)type I error; 5) type II error;
6)rejection region; 7)test value.
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