Assume the monthly rainfall in a certain state is normally distributed with a mean of 2.85 inches and a standard deviation of .2 inches:
a) probability that in any given month there is between 2.7 and 2.9 inches of rainfall
b) probability that in any given month there is more than 2.9 inches of rainfall
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
To find the probabilities in this scenario, we can use the standard normal distribution table or the z-score formula. The z-score formula allows us to convert the raw data into standardized values that can be easily compared.
a) To find the probability that in any given month there is between 2.7 and 2.9 inches of rainfall, we need to calculate the area under the normal distribution curve between these two values.
Step 1: Calculate the z-scores for each value using the formula:
z = (x - μ) / σ
Where:
x = value (2.7 and 2.9)
μ = mean (2.85)
σ = standard deviation (0.2)
For 2.7 inches:
z1 = (2.7 - 2.85) / 0.2 = -0.75
For 2.9 inches:
z2 = (2.9 - 2.85) / 0.2 = 0.75
Step 2: Use a standard normal distribution table or a calculator to find the corresponding probabilities for z1 and z2.
P(z < -0.75) = 0.2266
P(z < 0.75) = 0.7734
Step 3: Calculate the probability that the rainfall is between 2.7 and 2.9 inches.
P(2.7 < x < 2.9) = P(z < 0.75) - P(z < -0.75)
P(2.7 < x < 2.9) = 0.7734 - 0.2266
P(2.7 < x < 2.9) = 0.5468
Therefore, the probability is approximately 0.5468 or 54.68%.
b) To find the probability that in any given month there is more than 2.9 inches of rainfall, we need to calculate the area under the normal distribution curve to the right of the value 2.9.
Step 1: Calculate the z-score using the formula:
z = (x - μ) / σ
Where:
x = value (2.9)
μ = mean (2.85)
σ = standard deviation (0.2)
z = (2.9 - 2.85) / 0.2 = 0.25
Step 2: Use a standard normal distribution table or a calculator to find the corresponding probability for the z-score.
P(z > 0.25) = 1 - P(z < 0.25)
P(z > 0.25) = 1 - 0.5987
P(z > 0.25) = 0.4013
Step 3: Calculate the probability that the rainfall is more than 2.9 inches.
P(x > 2.9) = P(z > 0.25)
P(x > 2.9) = 0.4013
Therefore, the probability is approximately 0.4013 or 40.13%.