課程名稱︰統計學一上
課程性質︰工管系必修
課程教師︰蔣明晃
開課系所︰工管系
考試時間︰2006.01.09
試題 :
1.Collisions of ships in a shipping channel follow a Poisson distribution with
μ=0.5 collision per year. Let Y denote the number of collision in a year and
let X denote the time interval between successive collisions.
(a)P(X>1)
(b)P(Y=0)
2.A cosmetics firm must purchase a bottle-filling machine and has narrowed the
choice to one of two machines(1,2). The purchased machine will be used to
fill 100 milliliter(ml) bottles of cologne. Both machines can control the
mean fill at any desired level, but the standard deviation of fill is 0.3 ml
for machine 1 and 0.1 ml for machine 2. The fill with each machine are
normally distributed.
(a)At what mean fill level should each machine be set so that 99.9% of filled
bottles contain 100 ml or more of cologne?
(b)Consider the mean fill levels obtained in (a). The cost of cologne is$0.10
per ml. The cost of using machine 1 to fill 1000 bottles is $50. The
corresponding cost for machine 2 is $80. Other production and operating
cost for the two machines are identical. Obtain for each machine the
expected total cost of machine usage and of the cologne for filling 1000
bottles. What is the expected total cost per bottle for each machine? On
this basis, which machine is the better choice?
3. A battery-powered safety light is required to operate 400 hours without
failing. The light fails when its battery pack fails. Two configurations of
the battery pack (1,2) are being considered. In configuration 1, the pack
consists of a single battery having a normal distributed opertional lifetime
with a mean of 450 hours and a standard deviation of 30 hours. In
configuration 2, the pack consists of two batteries, designed so that the
second battery begins to operate when the first battery fails. Each of these
batteries has a normal distributed operational lifetime with a mean of 220
hours and a standard deviation of 15 hours. The operational lifetimes of the
two batteries are independent. Let T1 and T2 denote the operational
lifetimes of battery packs with configurations 1 and 2, respectively.
(a)Specify the nature of the probability distribution of T2.
(b)For each configuration, what is the probability that the battery pack
will have an operational lifetime of 400 hours or longer? According to
these probabilities, which is the better configuration?
(c)In a single trial of two battery packs, one with each configuration, what
is the probability that the configuration 1 pack will have a longer
operational lifetime than the configuration 2 pack? Assume that the lives
of the two packs are independent. Which appears to be the better
configuration based on this result?[Hint:Consider the difference of T1
and T2]
4. Consider the following probability distribution for an infinite population:
x 0 2
----------------------
P(x) 0.5 0.5
(a)What is the standard deviation for this population?
(b)For a simple random sample of n=2 from this population, construct (1) the
sampling distribution of s^2 and (2)the sampling distribution of s.
(c)Verify the s^2 is an unbiased estimate of σ^2, but s is a biased estimate
of σ. What is the amount of bias in s here?
5. In a study of the effect of nutrition on work efficiency, a random sample of
16 workers was asked to follow a rigid dietary regimen. In one part of this
study, the blood sugar level of X of each sample worker was measured two
hours after breakfast. The results(in milligrams of sugar per 100 cubic
centimeters of bloods) were x bar = 112.8 mg and s= 9.6 mg. Blood sugar
levels may be considered to be normally distributed.
(a)Construct a 95% confidence interval for the mean blood sugar level under
this regimen.
(b)The dietary regimen is intended to yield a mean blood sugar level of 110
mg of sugar per 100 cubic centimeters of blood. It is desired to test
whether this target is being met or not, controlling the α risk at 0.05
when μ=110
(1)Conduct the test. State the alternatives, the decision rule, the value
of the standardized test statistic, and the conclusion.
(2)Obtain the bound of the P-value.
(3)Use 95% confidence interval to test whether the mean blood sugar is
110 or not. What is the implied α risk of this test?
6. A trade agreement governing the movement of agricultural products between
two countries stipulates that the mean weight of boxes of butter must be
25.00 kilograms. A large shipment of boxes of butter is used to be tested to
determine whether it meets this requirement by selecting a random sample of
49 boxes and using the following decision rule:
If 24.95<=x bar<=25.05, conclude μ=25.00.
If x bar<24.95 or x bar >25.05, conclude μ≠25.00
The standard deviation of weights of boxes of butter is σ=0.15 kilogram.
(a)Calculate the rejection probability at μ=24.90, 25.00, 25.10 for this
decision rule, and complete the following table:
μ: 24.90 24,95 25.00 25.05 25.10
------------------------------------------------
P(H1;μ) ˍˍˍ 0.500 ˍˍˍ 0.500 ˍˍˍ
(b)Sketch the rejection probability curve for this decision rule.
(c)What is the α risk at μ=25.00 for this decision rule? What is the error
risk at μ=25.10? Does the latter relate to a Type Ⅰ or a Type Ⅱ error?
(d)It is desired to control the α risk at 0.05 when μ=25.00 and the β
risk at 0.05 when μ=24.95. Using σ=0.15 as the planning value, obtain
the required sample size.
(e)A random sample of the size determined at (d) has been selected, and the
number results are x bar=25.03 and s=0.183. Conduct the required test,
starting the decision rule, the value of standardized test statistic, and
the conclusion.
(f)The value of the sample standard deviation s is larger than the planning
value of σ used in obtaining the sample size. What does this difference
suggest about the magnitude of the actual β risk for the test at
μ=24.95 in relation to the target β risk of 0.05?
7. Nationally, the proportion of marriages ending in divorce or annulment that
involve no children is 0.40. In one state, a random sample of 1000 divorce
or annulments showed that 437 involved no children. It is desired to test
whether the state proportion is equal to 0.40. The α risk is to be
controlled at 0.01 when p=0.40
(a)Conduct the appropriate test. State the alternatives, the decision rule,
the value of the test statistic, and the conclusion.
(b)What is a Type Ⅱ error in this test situation? Is it possible that the
conclusion in (a) has resulted in a Type Ⅱ error?
(c)Calculate the P-value of the test.
(d)Obtain the rejection probability for each of the following values of p:
(1)0.53 (2)0.40 (3)0.45
(e)Plot the rejection probability curve for the decision rule in (a). What
is the β risk for this decision rule at p=0.45?
8. A Pharmaceutical firm produces tablets that need to contain a consistent
amount of active ingredient. For a random sample of 41 tablets just taken
from the production process, the standard deviation of the amounts of active
ingredient was s=1.09 milligrams. The amounts of active ingredient in
tablets are normally distributed.
(a)Construct a 95% confidence interval for the process variance of amounts
of active ingredient in tablets.
(b)Convert the confidence interval in (a) into a 95% confidence interval for
the process standard deviation of amounts of active ingredient in tablets.
(c)Quality standards require that the process variance for the amount of
active ingredient be 1.10 or less. Test the alternatives controlling the
α risk at 0.025 when σ^2 =1.10. State the decision rule, the value of
the test statistic, and the conclusion.
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