The annual precipitation for one city is normally distributed with a mean of 390 inches and a standard deviation of 3.2 inches. Fill in the blanks.
In 95% of the years, the precipitation in this city is between ___ and ___ inches.
You can play around with Z table stuff at
http://davidmlane.com/normal.html
go to
http://davidmlane.com/normal.html
click on "value from an area"
in Area --- enter .95
Mean --- 390
SD --- 3.2
click on the "between" button to get from 383.727 to 396.273
To find the range within which the precipitation would fall in 95% of the years, we need to calculate the z-scores corresponding to the lower and upper limits.
We know that the mean (μ) is 390 inches and the standard deviation (σ) is 3.2 inches.
For a 95% confidence interval, we can use a z-score of ±1.96 (which corresponds to a cumulative probability of 0.025 at each tail of the distribution).
The lower limit of the interval can be found by subtracting 1.96 times the standard deviation from the mean:
Lower limit = μ - (z * σ)
= 390 - (1.96 * 3.2)
The upper limit of the interval can be found by adding 1.96 times the standard deviation to the mean:
Upper limit = μ + (z * σ)
= 390 + (1.96 * 3.2)
Calculating these values, we get:
Lower limit = 390 - (1.96 * 3.2) ≈ 383.232 inches
Upper limit = 390 + (1.96 * 3.2) ≈ 396.768 inches
Therefore, in 95% of the years, the precipitation in this city is between approximately 383.232 and 396.768 inches.
To find the range of precipitation in which it falls for 95% of the years, we need to use the concept of z-scores and the standard normal distribution.
First, we need to find the z-score corresponding to the desired confidence level of 95%. We can do this using the z-table or a statistical calculator.
The z-score for a 95% confidence level (where 95% of the data falls within) is 1.96.
Next, we can use the formula for converting a z-score to an actual value in a normal distribution:
x = μ + (z * σ),
where x represents the actual value, μ is the mean, z is the z-score, and σ is the standard deviation.
Using this formula, we can find the lower and upper limits for the precipitation range.
Lower Limit:
x = 390 + (1.96 * 3.2)
x = 390 + 6.272
x ≈ 396.27 inches
Upper Limit:
x = 390 - (1.96 * 3.2)
x = 390 - 6.272
x ≈ 383.73 inches
Therefore, in 95% of the years, the precipitation in this city is between approximately 383.73 and 396.27 inches.