The size of fish is very important to commercial fishing a...

The size of fish is very important to commercial fishing. A study conducted in 2012 found the length of Atlantic cod caught in nets in Karlskrona to have a mean of 49.9 cm and a standard deviation of 3.74 cm. Round the probabilities to four decimal places. a) State the random variable. Select an answer: rv X = the length of a randomly selected Atlantic cod rv X = the mean length of a sample of Atlantic cod rv X = the mean length of all Atlantic cod rv X = length is normally distributed rv X = a randomly selected Atlantic cod b) Find the probability that a randomly selected Atlantic cod has a length of 47.08 cm or more. c) Find the probability that a randomly selected Atlantic cod has a length of 50.48 cm or less. d) Find the probability that a randomly selected Atlantic cod has a length between 47.08 and 50.48 cm. e) Find the probability that a randomly selected Atlantic cod has a length that is at least 62.99 cm.

Transcript

00:02 In this problem, x can be taken as length of a randomly selected atlantic code.

00:33 So for the first question, option a is the answer because this is a random variable and random variable x equal to length of a randomly selected atlantic code because length can be termed as a random variable.

01:15 Now we want to know the probability of a randomly selected atlantic code has length greater than or equal to 47 .08 or we can say it is more than 47 .08.

01:31 So the probability statement will be this.

01:35 And in order to do this, we can just assume that this x follows n mu sigma.

01:42 Since definitely the number of atlantic codes will be large, and here mu is equal to 49 .9 centimeters is available to us.

01:56 And also sigma is equal to 3 .74 which is also available.

02:02 This is the mean and standard deviation of the length.

02:05 So the standardization procedure here is xz equal to x minus mu by sigma.

02:15 This follows a standard normal distribution with the n01.

02:20 1.

02:21 Now we can do the problem using the standardization procedure as defined here.

02:27 So probability of x greater than or equal to 0 .0a can be standardized like this.

02:34 Probability of x can be standardized exactly and this term that is 47 .08 becomes 47 .08 minus mu.

02:46 Mu is here 49 .9 and which is divided by the standard deviation 3 .74.

02:54 So we can find the set score corresponding to 47 .08.

03:02 Now, the set score corresponding to 47 .08 is, is better than or equal to minus 0 .7540.

03:14 From the normal curve we can see that this is the curve and this minus point minus 7 540 will be somewhere over here the range is minus infinity to infinity and this is the required area but we are using cumulative tables standard cumulative tables so from tables this value will be available so we can calculate this using 1 minus probability of z less than minus .7540.

04:03 And from excel tables we have this is equal to 0 .22 by 4.

04:11 The excel tables are given here and the value is equal to 0 .7746.

04:20 So probability of x greater than or equal to 47 .08 is equal to 0 .7746...

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