a) The percentage of years that will have an annual rainfall of less than 43 inches is 67.4%.
b) The percentage of years that will have an annual rainfall of more than 38 inches is 67.4%.
c) The percentage of years that will have an annual rainfall of between 37 inches and 41 inches is 27.0%.
In Mathematics and Statistics, the z-score of a given sample size or data set can be calculated by using the following formula:
Z-score, z = (X - μ)/σ
Where:
- σ represents the standard deviation.
- X represents the sample score.
- μ represents the mean score.
Part a.
First of all, we would standardize the variable X by subtracting the mean and dividing by the standard deviation as follows;
Z-score, z = (43 - 40.5)/5.6
Z-score, z = 0.45
Based on the standardized normal distribution table, the required probability is given by:
P(X < 43) = 1 - P(x > Z)
P(X < 43) = 1 - 0.3264
Percentage = 0.6736 × 100
Percentage = 67.4%
Part b.
Z-score, z = (38 - 40.5)/5.6
Z-score, z = -0.45
Based on the standardized normal distribution table, the required probability is given by:
P(X > 38) = 1 - P(x < Z)
P(X > 38) = 1 - 0.3264
Percentage = 0.6736 × 100
Percentage = 67.4%
Part c.
Z-score, z = (37 - 40.5)/5.6
Z-score, z = -0.625
Z-score, z = (41 - 40.5)/5.6
Z-score, z = 0.089
P(37 ≤ x ≤ 41) = P(-0.625 ≤ z ≤ 0.089)
P(37 ≤ x ≤ 41) = P(z ≤ 0.089) - P(z ≤ -0.625)
P(37 ≤ x ≤ 41) = 0.5355 - 0.2660
P(37 ≤ x ≤ 41) = 0.2695 × 100 = 27.0%
Complete Question:
The annual rainfall in a certain region is approximately normally distributed with mean 40.5 inches and standard deviation 5.6 inches. Round answers to the nearest tenth of a percent.
a) What percentage of years will have an annual rainfall of less than 43 inches?
b) What percentage of years will have an annual rainfall of more than 38 inches?
c) What percentage of years will have an annual rainfall of between 37 inches and 41 inches?