Answer:
Step-by-step explanation:
<pGreetings!
a. The variable X represents the height of a Goomba, which follows a normal distribution with a mean of μ= 12 inches and a standard deviation of δ= 6 inches.
To find the probability that a Goomba picked at random has a height between 13 and 15 inches, you express it as:
P(13≤X≤15)
Considering that standard normal probability tables provide cumulative values, you can express this range as the cumulative probability up to 15 minus the cumulative probability up to 13. You'll first need to standardize these variable heights to obtain corresponding Z values:
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 we have Y as the variable indicating the height of a Koopa Troopa. This variable also follows a normal distribution, with a mean μ= 15 inches and a standard deviation δ=3 inches.
The query concerns the probability that a Koopa Troopa stands taller than 75% of Goombas.
First step:
You need to determine the height of a randomly chosen Koopa Troopa that exceeds 75% of the Goomba population.
This entails determining the value of X corresponding to the limit below which 75% of the population falls, denoted by:
P(X ≤ b)= 0.75
Step 2:
Search the standard normal distribution for the Z value that has 0.75 beneath it:
Next, you will reverse the standardization to solve for "b"
Z= (b - μ)/δ
b= (Z*δ)+μ
b= (0.674*6)+12
b= 16.044 inches
Step 3:
With the height that identifies a Koopa Troopa taller than 75% of the Goomba population determined, compute the probability of selecting that Koopa Troopa:
P(Y≤16.044)
This time, utilize the Koopa’s average height and standard deviation to find the probability:
P(Z≤(16.044-15)/3)
P(Z≤0.348)= 0.636
The likelihood of randomly selecting a Koopa Troopa that is taller than 75% of Goombas is 63.6%
I hope this information is useful!
