The yearly rainfall in Vancouver, B.C. follows a normal distribution with standard deviation 172 mm. In 20% of years, the city gets over 1200 mm of rain. What is the mean annual rainfall in Vancouver (in mm)?
The answer is 1055.5 mm....not sure how to get this answer.
why is the z-score 0.84
took me way too long but i got it:
1 - .20 = .8000
*Did this because you are looking for P(x>1200) therefore, you are looking at for a right side probability, z table only give you left.
Find .8000 INSIDE z chart, the closest number is .7995 = .84
x - (z*std.dev) = 1200 - (.84*172) = 1055.5
To find the mean annual rainfall in Vancouver, we can use the properties of a normal distribution.
First, let's understand what the problem is telling us:
- The yearly rainfall in Vancouver follows a normal distribution.
- The standard deviation is given as 172 mm.
- In 20% of years, the city gets over 1200 mm of rain.
We need to find the mean annual rainfall.
To solve this, we can use the concept of Z-scores. A Z-score measures the number of standard deviations an observation or data point is from the mean.
Step 1: Find the Z-score corresponding to 20% of years above 1200 mm rainfall.
In a standard normal distribution, the Z-score corresponding to the 20th percentile is -0.84 (you can find this value using a Z-table or a statistical software). This means 20% of the data lies to the right of -0.84 on the standard normal distribution curve.
Step 2: Use the Z-score to find the corresponding raw value (1200 mm) in the original rainfall distribution.
The formula to convert a Z-score to a raw value is:
Z = (X - μ) / σ
where Z is the Z-score, X is the raw value, μ is the mean, and σ is the standard deviation.
In this case, we know the Z-score (-0.84), the raw value (1200 mm), and the standard deviation (172 mm). We need to solve for the mean (μ).
-0.84 = (1200 - μ) / 172
Step 3: Solve for the mean (μ).
Multiply both sides of the equation by 172:
-0.84 * 172 = 1200 - μ
Simplify:
-144.48 = 1200 - μ
Rearrange the equation:
μ = 1200 - (-144.48) = 1200 + 144.48 = 1344.48 mm
So, the mean annual rainfall in Vancouver is approximately 1344.48 mm.
It seems there might be an error in the provided answer of 1055.5 mm. The correct answer should be 1344.48 mm. Please verify the information and calculations to double-check the result.
a z-score table will give the number of s.d. above the mean for the 20% figure
the z-score is .84
so 1200 mm is .84 s.d. above the mean
1200 - (.84 * 172) = ?