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(MATH144)[2009](s)final~1688^_10458.pdf
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Math 144: Final examination 2009/2010, Dec 16, 2009.

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All answers should be either exact or correct to 4 decimal places.
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1. [10 marks] Poker dice is played by simultaneously rolling 5 fair dice. Find the following probabilities:
(a) P{no two alike}, (b) P{one pair}, (c) P{two pair}, (d) P{.ve alike}.
2. The weight (in tons) of a vehicle crossing a certain bridge is found to have a density function shown below.
(a)
[2 marks] Write down the equation for the increasing line in the picture and that for the decreasing line.

(b)
[2 marks] What proportion of a vehicle that crosses the bridge weighs more than 1.5 tons?

(c)
[2 marks] Find the mean and standard deviation of the weight of a vehicle?

(d)
[2 marks] If there are 50 vehicles on the bridge, what is the normally-approximated probability that their combined weight exceeds 200 tons?

(e)
[4 marks] If there are 50 vehicles on the bridge, what is the probability that the heaviest ve-hicle exceeds 9.7 tons? [Hint: the weights of these 50 vehicles are independent and identically distributed.]


3. (a) [6 marks] Suppose that a surveyor is trying to determine the area of a rectangular .eld, in which the measured length X and the measured width Y are independent random variables that .uctuate widely about the true values, according to the following probability distributions:
X 8 10 11
Probability 0.25 0.25 0.5

Y 4 6
Probability 0.5 0.5

The calculated area A = XY of course is a random variable, and is used to estimate the true area. If the true length and width are 10 and 5, respectively, are X, Y and A unbiased?
(b) [3 marks] Let X1, ,Xn be a random sample from a normal population N(0,2), where 2 is
1
unknown. Let W = .n Xi 2 . Prove that W is an unbiased estimator of 2 .
ni=1
1
4. A recent article in the British journal Lancet reports that babies who were fed mothers milk tended to have a higher IQ than formula-fed babies. Suppose that two groups of babies are compared, a group fed mothers milk and a group fed formula. The resulting IQ scores are as follows:
Mother: 121, 105, 111, 119, 108, 101, 90, 131, 106, 112 Formula: 101, 110, 107, 98, 89, 103, 86, 117, 113, 87
Assume that the IQ scores are normally distributed with population mean 1 and population variance 2 for mother-fed babies; population mean 2 and population variance 2 for formula-fed babies.
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(a)
[4 marks] Construct a 95% con.dence interval for the mean IQ score for each group.

(b)
[4 marks] Assume that the two population variances are equal. Construct a 95% con.dence interval for the di.erence 1 . 2 between the mean IQ scores. What can be concluded from this con.dence interval?

(c)
[3 marks] At a 0.05 level of signi.cance, te