Module 2 - SGTA 3 - solution version 4 PDF

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Introduction to Distributions In this exercise we will: • Calculate probabilities for a Normal distribution. Normal Distribution The Normal distribution is the most important distribution for continuous numerical variables. It is used as a model for many measurements that occur in nature and its use is further extended as a result of the Central Limit Theorem. The Normal distribution has a symmetric “bell” shape. It is described by two parameters; the centre of the distribution, μ, and its standard deviation, σ, a measure of spread. There are an infinite number of Normal distributions. Each one has the same “bell” shape but the values of μ, the population mean, and σ, the population standard deviation, vary. In lectures we learned about the Normal distribution. If we have a population with a Normal distribution, we can calculate probabilities for values of the random variable, ฀฀, using probTool(MQ).

Solving Probability Problems The following steps are a useful method for solving probability problems: 1. 2. 3. 4.

Draw a diagram. Shade in the required area. Find relevant probabilities. Write a sentence which summarises your findings.

Introduction to Distributions

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Probabilities for the Normal Distribution

Effective management of global fisheries and aquaculture are essential to the future viability of fish species and the continued availability of fish as a resource. Management of fish species includes ensuring that only fish of legal length are caught. Larger fish must also be left in the population as larger female fish lay many more eggs than smaller female fish. The Normal distribution has been selected to model the length of fish in a population. We have found that this population of fish is approximately normally distributed with an average length of 34cm, with a standard deviation of 11cm. Use the method for solving probability problems in conjunction with probTool(MQ) to answer the following questions:

1. What is the probability that the length of a fish in this population is less than 25 cm?

฀฀฀฀฀฀฀฀ (฀ ฀ < ฀฀฀฀) = ฀฀. ฀฀฀฀฀฀฀฀ There is a 21% probability that a fish from this population is less than 25cm.

2. What is the probability that the length of a fish in this population is more than 45 cm?

฀฀฀฀฀฀฀฀ (฀฀ > ฀฀฀฀) = ฀฀. ฀฀฀฀฀฀฀฀ There is a 16% probability that a fish from this population is greater than 45cm. Introduction to Distributions

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3. A fish from this population must be between 28cm and 52cm long to be legal to catch. What is the probability that a fish from this population is within the legal limit for length? Distribution Plot Normal, Mean=34, StDev=11 0.04

0.6564

Density

0.03

0.02

0.01

0.00

28

34

52

X

฀฀฀฀฀฀฀฀ (฀฀฀฀ < ฀฀ < ฀฀฀฀) = ฀฀. ฀฀฀฀฀฀฀฀ There is a 66% probability that the length of a fish from this population is within the legal limit to catch.

4. Any fish whose length is in the top 5% need to be released. What is the length of a fish for which only 5% of fish will be greater than that length? Distribution Plot Normal, Mean=34, StDev=11 0.04

Density

0.03

0.9500

0.02

0.01

0.00

34

52.093

Y

฀ ฀ = ฀฀฀฀. ฀฀฀฀ Fish from this population which are longer than 52cm are in the top 5% of fish ` by length. Any fish which are longer than 52cm will need to be released.

Introduction to Distributions

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5. If 10 fish are caught from this population, how many would be expected to be greater than 45 cm long? There is a 16% probability that a fish from this population is greater than 45cm. ฀฀฀฀฀฀฀฀ (฀ ฀ > ฀฀฀฀) = ฀฀. ฀฀฀฀฀฀฀฀฀฀฀฀ The expected number of fish = ฀฀ × ฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀฀ = ฀฀฀฀ × ฀฀. ฀฀฀฀฀฀฀฀฀฀฀฀ = ฀฀. ฀฀฀฀ From a catch of 10 fish, we would expect to catch 1 or 2 fish greater than 45cm.

6. For any variable, y, from a Normal distribution, we can calculate a z-score. The z-score is the number of standard deviations that a value, y, is away from its population mean, μ. The z-score converts the random variable, y, to a standardised random variable, z, which has a mean of 0 and a standard deviation of 1. A z-score is calculated using the formula: ฀ ฀ =

฀฀−฀฀ ฀ ฀

What is the z-score for a fish whose length is 56cm?

฀฀=

฀฀−฀฀ ฀

฀

฀฀฀฀−฀฀฀฀

= ฀฀฀฀ = ฀฀

Using this z-score, what is the probability that a fish is more than 56cm in length? Distribution Plot Normal, Mean=0, StDev=1 0.4

Density

0.3

0.2

0.1

0.02275 0.0

0

2

z

฀฀฀฀฀฀฀฀ (฀ ฀ > ฀฀) = ฀฀. ฀฀฀฀฀฀฀฀฀฀ There is a 2% probability that a fish from this population is over 56cm in length.

Introduction to Distributions

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