The growth rate of abalone can be monitored using a statistic known as the speci
ID: 2922901 • Letter: T
Question
The growth rate of abalone can be monitored using a statistic known as the specific growth rate (SGR). If the growth rate for abalone in a tank is too low, it is not economic to spend money and use resources to continue to grow the abalone. In the early stages, a sample of abalone is monitored to provide the information on which a decision will be based. There are two options:
Option 1. discard this batch and start the process again with a new batch of abalone because the growth rate is to0 low;
Option 2. allow the present stock to continue growing.
Decision-making rule. To make a decision, a sample of ten abalone is tagged, the mean SGR is determined for the tagged shellfish, and a decision is made based on the following rule:
Choose Option 1 if the mean SGR is less than 0.54.
Choose Option 2 if the mean SGR is over 0.54.
Probabilities of choosing different options. From past observation, it is known that the pattern of SGR among individual abalone is well approximated by a Normal distribution with standard deviation 0.2
a) For the indicated mean values for SGR in the table below, complete the table to establish the probabilities that the decision rule stated above will lead to the selection of Option 1 or Option 2.
Probabilities of selecting different options based on mean SGR for ten abalone for different mean levels of SGR for the tank
Mean SGR for tank 0.60 0.64 0.68 0.72
Probability
Option 1
Option 2
Proportion and number above the minimum acceptable SGR level. The minimum acceptable level of SGR for an abalone to reach a commercially acceptable size at the end of the growing period is 0.55.
b) Let the random variable Y be the SGR of an individual abalone. For each value of the 'mean SGR for the tank', compute the proportion of abalone in the tank which have an SGR level above the minimum acceptable level of 0.55, i.e. P (Y>0.55).
Mean SGR for tank 0.60 0.64 0.68 0.72
P(Y>0.55)
c) If the mean SGR in the tank is 0.64, what distribution best describes the number of abalone in a batch of 100 that achieve the minimal acceptable growth rate?
d) Is it valid to approximate the distribution above with a normal distribution? If so what normal distribution would you use?
Chances of making a profit. A profit cannot be made under Option 1 since the process is stopped before the abalone mature. Under Option 2, for a batch of 100 abalone to return a profit, at least seventy percent of the abalone must have SGR levels above the minimum acceptable value of 0.55.
Chances of making a profit. A profit cannot be made under Option 1 since the process is stopped before the abalone mature. Under Option 2, for a batch of 100 abalone to return a profit, at least seventy percent of the abalone must have SGR levels above the minimum acceptable value of 0.55.
Probabilities of making a profit under option 2 based on mean SGR for ten abalone for different mean levels of SGR for the tank
Mean SGR for tank 0.60 0.64 0.68 0.72
70 or more acceptable
e) For batches of 100 abalone, compute the probabilities that the number of abalone is at least equal to 70 and use these values to complete the table below.
f) Assuming that the mean SGR is 0.64 what is the overall probability of making a profit?
g) Do you think that the decision rule used by the company is reasonable?
Explanation / Answer
a) It is given that SGR among individual abalone follows a Normal distribution with standard deviation 0.2.
Hence the mean of a sample of 10 abalone follows a normal distribution with the same mean and
has a standard deviation, s = 0.2/sqrt(10) = 0.2/3.1623 = 0.0632
Hence probabilities of Option 1 (mean SGR < 0.54) and Option 2( mean SGR > 0.54) for various values of population mean SGR can be easily found using the Normal Distribution Function in Excel:
Mean SGR for tank 0.60 0.64 0.68 0.72
Probability of Option 1 0.1712 0.0568 0.0134 0.0022
Probability of Option 2 0.8288 0.9432 0.9866 0.9978
b) SGR of individual abalones follows a normal distribution with sd = 0.2. Hence P(Y>0.55) can again be easily found out using the NORMDIST function in Excel:
Mean SGR for tank 0.60 0.64 0.68 0.72
P (Y > 0.55) 0.5987 0.6736 0.7422 0.8023
c) If the mean SGR in the tank is 0.64, the number of abalone in a batch of 100, that achieve the minimum acceptable growth rate of 0.55, will follow a binomial distribution with n=100 and p = 0.6736.
d) Yes, it is valid to approximate the above binomial distribution with a normal distribution because of the Central Limit Theorem, since the sample size of 100 is reasonably large.
The mean of this normal distribution will be np = 100 X 0.6736 = 67.36
The standard deviation will be sqrt (np(1-p)) = sqrt(100 X 0.6736 X 0.3264) = 4.6890
Please note I have already answered 4 sub-parts, a, b, c and d as per the answering guidelines. Hence. I am skipping sub-parts e f and g.
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