Question

At Pike Place Fish Market in Seattle, customers can purchase a variety of different types of seafood. One type of seafood sold at the market is rainbow trout. If we examine the distribution of the weights the rainbow trout sold at Pike Place Fish Market, we find the distribution is Normal, with a mean of 3.6 pounds and a standard deviation of 0.8 pounds. You would like to purchase a rainbow trout that weighs more than 90% of all other rainbow trout sold at the market. This means your rainbow trout would need to weigh a minimum of A. 3.68 pounds. B. 4.40 pounds. C. 4.64 pounds. D. 4.72 pounds. E. 5.52 pounds.

Answer 1

Answer:

$z=1.28$

And if we solve for a we got

$a=3.6 +1.28*0.8=4.624$

So the value of height that separates the bottom 90% of data from the top 10% is 4.624.

So then the best answer for this case would be:

 C. 4.64

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the weights of a population, and for this case we know the distribution for X is given by:

$X \sim N(3.6,0.8)$  

Where $\mu=3.6$ and $\sigma=0.8$

For this part we want to find a value a, such that we satisfy this condition:

$P(X>a)=0.1$   (a)

$P(X$   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.9 of the area on the left and 0.1 of the area on the right it's z=1.28. On this case P(Z<1.28)=0.9 and P(z>1.28)=0.1

If we use condition (b) from previous we have this:

$P(X$  

$P(z$

But we know which value of z satisfy the previous equation so then we can do this:

$z=1.28$

And if we solve for a we got

$a=3.6 +1.28*0.8=4.624$

So the value of height that separates the bottom 90% of data from the top 10% is 4.624.

So then the best answer for this case would be:

 C. 4.64