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Unformatted text preview: BES Tutorial Sample Solutions, S1/11
WEEK 7 TUTORIAL EXERCISES (To be discussed in the week starting
April 11)
1. From several years’ records, a fish market manager has determined that the
weight of deep sea bream sold in the market (X) is approximately normally
distributed with a mean of 420 grams and a standard deviation of 80 grams.
Assuming this distribution will remain unchanged in the future, calculate
the expected proportions of deep sea bream sold over the next year
weighing
(a) between 300 and 400 grams.
300
300
400 80
1.5 0
0.4332
0.3345
(b) between 300 and 500 grams.
300
300
500 420 (c) more than 600 grams.
600 0.25 500 420
1.5
0.3413 600 420
80 1 0 420
80 2.25
0.5
0
0.5 0.4878
0.0122 420
80 0.25
1.5
0
0.0987 80
1.5 0
0.4332
0.7745 400 2.25 1 2. In a certain large city, household annual incomes are considered
approximately normally distributed with a mean of $40,000 and a standard
deviation of $6,000. What proportion of households in the city have an
annual income over $30,000? If a random sample of 60 households were
selected, how many of these households would we expect to have annual
incomes between $35,000 and $45,000?
~
30000 40000, 6000 ) 30000 40000
6000
1.67
0.5
0
1.67
0.5 0.4525
0.9525 So 95.25% of households in the city have annual incomes greater than $30,000. 35000 40000
45000 40000
35000
45000
6000
6000
0.83
0.83
20
0.83
2 0.2967
0.5934 Therefore we expect 0.5934(60)≈36 households in the sample to have annual incomes between $35,000 and $45,000. 3. In a certain city it is estimated that 40% of households have access to the
internet. A company wishing to sell services to internet users randomly
chooses 150 households in the city and sends them advertising material.
For the households contacted:
(a) Calculate the probability that less than 60 households have internet
access?
Let X be the number of households contacted that have internet access. Then assume X is a binomial random variable with n=150 and p=0.4. Because n is large we can use the normal approximation to the binomial where: 150 0.4 60 1
150 0.4 0.6 36 Thus incorporating the continuity correction we need to find: 60
59
59.5 59.5 60 6
0.083 0.5
0
0.083 0.5 0.0319 0.4681 (b) Calculate the probability that between 50 and 100 (inclusive)
households have internet access?
50 (c) 100 49.5
100.5 49.5 60
100.5 60 6
6
1.75
6.75 0
1.75
0
6.75 0.4599 0.5 0.9599 Calculate the probability that more than 50 households have internet
access?
51 0.5 50.5 50.5 60 6
1.583 0.4429 0.9429 (d) There is a probability of 0.9 that the number of households with internet
access equals or exceeds what value?
0.9
0.5 0 0.5 60
0.9 6
60.5
0.4 6
60.5
1.28
52.82 6 0.9 There is a 90% chance that the number of households with internet access is 52 or more. 4. The manufacturer of a particular handmade article takes place in two
stages. The time taken for the first stage is approximately normally
distributed with a mean of 30 minutes and a standard deviation of 4
minutes. The time taken for the second stage is also approximately
normally distributed, but with a mean of 10 minutes and a standard
deviation of 3 minutes. The times to complete the two stages of production
are independently distributed. Note that the sum of two normally
distributed random variables is also normally distributed.
Let X1 be the time taken for the first stage and X2 the time taken for the second stage. Then Y = X1 + X2 is the time taken to complete an article. (a) What are the mean and standard deviation of the total time to
manufacture the article?
30 5 ~ 3
4
40,25 10 40 25 Note that independence is used in the calculation of the Var(Y) (b) What is the probability of finishing an article in less than 35 minutes?
35 35 40
5 1
0.5
0
0.5 0.3413
0.1587 1 (c) What proportion of articles will be completed in 3545 minutes?
35 45 35
1 45 40
5 2
0
0.6826 40
5 1
1 5. What is the 25th percentile of the normal distribution N(10, 9)? Let x be the required percentile. First find z, the 25th percentile of a standard normal. 0.25 0.25 0
0.5 .25 0.25 0.67
0.67 ~ 10,9
25
: 10
0.67 3
7.99 ...
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This note was uploaded on 05/24/2011 for the course ECON 1293 taught by Professor Denzilgfiebig during the Three '11 term at University of New South Wales.
 Three '11
 DenzilGFiebig

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