Maths Paper Notingham Uni
MT230
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?
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?