# Maths Paper Notingham Uni

**Topics:**Standard deviation, Normal distribution, Variance

**Pages:**2 (333 words)

**Published:**March 6, 2013

2010/2011

Exercise 1 (To be returned on Tuesday, October 5)

Question 1

A mechanical jar ﬁller is used to ﬁll jars with coﬀee. The ﬁller is set so that the mean jar ﬁll is 205 grams. The standard deviation of the jar ﬁlls is 2.5 grams. If the population of jar ﬁlls is normally distributed, what percentage of jar ﬁlls will be (i) greater than 202.5 grams, (ii) between 201.25 and 208.75 grams, and (iii) greater than 212.5 grams? If coﬀee jars have a capacity of 200 grams, (iv) what percentage are underﬁlled by the jar ﬁller and, (v) how would you reset the mean jar ﬁll so that only 0.4% are underﬁlled? Question 2

(i) If random variables Y1 , Y2 , · · · , Yn are independent with a common mean µ but 2

2

2

have variances σ1 , σ2 , · · · , σn (which need not be the same) show that the sample mean ¯

Y is an unbiased estimator of µ with standard deviation

1

n

1

2

n

2

σi

.

i=1

(ii) A student takes independent measurements of an instrument but, owing to fatigue, 2

the variance of his measurements increases linearly, i.e., σr = σ 2 + rc where σ 2 > 0 ¯

c > 0 (r = 1, · · · , n). Use the result of (i) to show that the variance of Y is never less 1

¯ is not a consistent estimator of µ in this case.

than 2 c. Hence show that Y

Question 3

Eight international athletes ran a 400 meter race at sea level (London, UK). A month later they raced at an altitude of 5220 feet above sea-level (Denver, USA). athlete

1

2

3

4

5

6

7

8

London UK 47.79 46.64 45.67 44.61 45.20 47.11 46.04 48.22

Denver USA 47.62 45.93 46.20 44.82 44.24 45.42 46.21 47.55

Would you say that the athletes ran quicker at 5000 feet?

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