iN manoa Valley on the island of Oahu Hawaii the annual rainfall averages 43.6 inches with a standard diviation 7.5 inches. for a given year, find the probability that the annual rainfall will be..(solve for a,b,and c) show work
a)more than 53 inches
b)less than 28 inches
c)between 28 and 53 inches
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.
To find the probability of a certain event occurring, we need to use the concept of standard deviation and z-scores. The z-score measures how many standard deviations a given value is from the mean. We can then use a standard normal distribution table or a calculator to find the corresponding probability.
First, let's find the z-scores for each scenario:
a) More than 53 inches:
To find the z-score, we can use the formula: z = (x - μ) / σ, where x is the given value, μ is the mean, and σ is the standard deviation.
In this case, x = 53, μ = 43.6, and σ = 7.5.
z = (53 - 43.6) / 7.5
z = 9.4 / 7.5
z = 1.253
b) Less than 28 inches:
Using the same formula, we have x = 28, μ = 43.6, and σ = 7.5.
z = (28 - 43.6) / 7.5
z = -15.6 / 7.5
z = -2.08
c) Between 28 and 53 inches:
To find the probability of being between two values, we need to find the area under the curve between the corresponding z-scores. In this case, we need to find the area between z = -2.08 and z = 1.253.
Using a standard normal distribution table or a calculator, we can find the probabilities:
P(a) = P(z > 1.253)
P(b) = P(z < -2.08)
P(c) = P(-2.08 < z < 1.253)
Now, let's find the probabilities using the z-scores:
a) P(a) = 1 - P(z < 1.253)
This calculation can be done using the standard normal distribution table or a calculator.
b) P(b) = P(z < -2.08)
Again, you can use the standard normal distribution table or a calculator.
c) P(c) = P(-2.08 < z < 1.253)
We need to find the probability between two z-scores.
By using these steps and the appropriate calculations, you can find the solutions for a), b), and c) according to the given mean and standard deviation.