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(MATH144)[2010](s)final~1487^_10051.pdf
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Math 144: Final examination (Spring 2010)

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This exam will be 2 hours long.

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Please show all your work so that you do not lose any marks.

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Unless otherwise speci.ed, numerical answers should be either exact or correct to 4 decimal places.


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1. (a) [5 marks] Consider the following hypothesis testing
H0 : = 120
H1 : = 100

Suppose that the test procedure is to reject H0 if X< 70.65, where X N(, 30). Then draw a picture and explain why we cannot control Type I and II errors simultaneously.
(b) [5 marks] The following are the weights, in decagrams, of 10 packages of grass seed distributed by a certain company:
46.4, 46.1, 45.8, 47.0, 46.1, 45.9, 45.8, 46.9, 45.2, and 46.0.
Assume that the package of grass seed distributed by this company has a normal distribution. Find a 95% con.dence interval for 1/.
2. 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
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mother-fed babies; population mean 2 and population variance 2 for formula-fed babies.
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(a)
[9 marks] Assume that the two population variances are NOT 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?

(b)
[3 marks] At a 0.05 level of signi.cance, test H0 : 1 .


= 2 + 25 against H1 : 1 = 2 + 25, when the two population variances are assumed to be unequal.
3. The volume of diamonds follows an exponential distribution with mean size 0.2 microliters. Diamonds that are small are used for drilling and other machinery, but diamonds that exceed 1.1 microliters in volume are considered jewellery worthy and their value increases sharply.
(a)
[3 marks] What is the probability that a randomly selected diamond will be considered jewellery worthy?

(b)
[4 marks] Jerry, the minor, collected 850 diamonds today. Their sizes are considered to be independent from one diamond to the next. What is the probability that exactly 5 of them are considered jewellery worthy?

(c)
[5 marks] Now if Jerry collects 50 diamonds, then what is the normally-approximated probability that their total volume will exceed 12 microliters?


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[8 marks] A model for the movement of a stock supposes that, if the present price of the stock is s, then . after one period . it will eithe