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Background The growth rate of abalone can be monitored using a statistic known ...

May 23, 2024

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Solution by Steps

step 1

We start by calculating the probabilities for the normal distribution with mean values given in the table and standard deviation

\sigma = 0.215

step 2

For each mean SGR value, we calculate the cumulative distribution function (CDF) at

0.56

to determine the probability of choosing Option 1

step 3

Using the CDF values, we can determine the probabilities for Options 2 and 3

step 4

For the mean SGR of

0.6

, the CDF at

0.56

0.4149

. Therefore, the probability of choosing Option 1 is

0.4149

step 5

The probability of choosing Option 2 is the difference between the CDF values at

0.60

and

0.56

step 6

The probability of choosing Option 3 is

1 - \text{CDF at } 0.60

step 7

Repeat steps 4-6 for each mean SGR value in the table

step 8

For the proportion of saleable abalone, we calculate

1 - \text{CDF at } 0.56

for each mean SGR value

step 9

For the binomial distribution, we use the proportion of saleable abalone to calculate the probabilities of having at least

50, 60, 70, 80,

and

90

acceptable abalone out of

100

step 10

For the profit probability, we use the binomial distribution results to determine the probability of making a profit under Options 2 and 3

Answer

The probabilities and proportions for each mean SGR value are calculated using the normal and binomial distributions.

Key Concept

Normal and Binomial Distributions

Explanation

The normal distribution is used to model the SGR values, and the binomial distribution is used to calculate the probabilities of having a certain number of acceptable abalone.