To celebrate their 30th birthdays, brothers Mario and Luigi of the Nintendo Mario video game franchise wish to study the distribution of heights of their mushroom enemies, the Goombas. Their reasoning is that shorter Goombas are easier to jump on. (Goombas die when Mario and Luigi jump on them.)(a) If we assume that the population of Goomba heights are normally distributed with mean 12 inches and standard deviation 6 inches, what is the probability that a randomly chosen goomba has a height between 13 and 15 inches?(b) Koopa Troopas, other enemies of Mario & Luigi, have a mean height of 15 inches with standard deviation 3 inches. What is the probability that a randomly chosen Koopa Troopa is taller than 75% of Goombas?

Step-by-step explanation:

Hello!

a. The variable X: height of a Goomba. This variable has a normal distribution with mean μ= 12 inches and a standard deviation δ= 6 inches.

You need to calculate the probability that one randomly chosen Goomba has a height between 13 and 15, symbolically:

P(13≤X≤15)

Since the probability tables for the standard normal show cumulative values, you can rewrite this interval as the accumulated probability until 15 minus the accumulated probability until 13. Then or before, you can standardize the values of the variable to obtain values of Z:

P(X≤15) - P(X≤13)

P(Z≤(15-12)/6) - P(Z≤(13-12)/6)

P(Z≤0.33) - P(Z≤0.17)= 0.62930 - 0.56749= 0.06181

b. Now the study variable is Y: height of a Koopa Troopa. This variable has also a normal distribution with mean μ= 15 inches and a standard deviation δ=3 inches.

The question asks for the probability of a Koopa Troopa that is taller than 75% of Goombas.

First step:

You have to reach the value of the height of a randomly selected Koopa Troopa that is taller than 75% of the Goombas.

This means that first, you have to work under X "height of a Goompa"

And you have to look for the value of X that leaves below 0.75 of the population, symbolically:

P(X ≤ b)= 0.75

Step 2:

You have to look in the standard normal distribution for the value of Z that has below it 0.75:

Next is to reverse the standardization and clear the value of "b"

Z= (b - μ)/δ

b= (Z*δ)+μ

b= (0.674*6)+12

b= 16.044 inches

Step 3:

After learning which is the height that corresponds to a Koopa Troopa higher than 75% of the Goombas population, you have to calculate the probability of randomly selecting said Koopa:

P(Y≤16.044)

This time you'll use the mean and standard deviation of the height of the Koopa's to calculate the probability:

P(Z≤(16.044-15)/3)

P(Z≤0.348)= 0.636

The probability of randomly selecting a Koopa Troopa higher than 75% of the Goomba population is 63.6%

I hope it helps!

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