The yearly rainfall in Regina, Saskatchewan follows a normal distribution with mean 384
mm and standard deviation o. In 10% of years, the city gets less than 320 mm of rain.
What is the standard deviation of the amount of annual rainfall in Regina?
To find the standard deviation of the amount of annual rainfall in Regina, we need to use the Z-score formula.
The Z-score formula is given by:
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the observed value
μ is the mean
σ is the standard deviation
In this case, we want to find the Z-score for the value X = 320 mm, which represents the amount of rainfall in the 10th percentile.
Since the normal distribution is symmetric, the area to the left of 320 mm is 10% (0.10), and the area to the right is 90% (0.90). We can look up the Z-score for the 90th percentile (Z0.90) in the Z-table or use a calculator.
Once we have the Z-score, we can rearrange the formula to solve for σ:
σ = (X - μ) / Z
Substituting the values into the formula:
σ = (320 - 384) / Z0.90
Now we need to find the Z-score for the 90th percentile. Looking up this value in the Z-table or using a calculator, we find Z0.90 ≈ 1.28.
Substituting this value back into the formula:
σ = (320 - 384) / 1.28
Calculating this, we get:
σ ≈ -50 / 1.28
Therefore, the standard deviation of the amount of annual rainfall in Regina is approximately -39.06 mm.