high temperatures in a certain city for the of the month of august follow a uniform distribution over the interval 73 degreef to 103 degree f find the high temperature which 90% of the august days exceeded
To find the high temperature that 90% of the August days exceeded, we can use the cumulative distribution function (CDF) of the uniform distribution.
In a uniform distribution, the CDF is given by the formula:
CDF(x) = (x - a) / (b - a)
where 'a' is the lower limit of the distribution (73°F) and 'b' is the upper limit of the distribution (103°F).
We want to find the value of 'x' for which CDF(x) = 0.9.
Plugging the values into the formula:
0.9 = (x - 73) / (103 - 73)
To solve for 'x', we multiply both sides of the equation by (103 - 73):
0.9 * (103 - 73) = x - 73
Simplifying:
0.9 * 30 = x - 73
27 = x - 73
Now, add 73 to both sides of the equation to solve for 'x':
27 + 73 = x
Final calculation:
x = 100
Therefore, the high temperature that 90% of the August days exceeded is 100°F.
To find the high temperature that 90% of the August days exceeded, we need to determine the value of the 90th percentile in the uniform distribution.
In a uniform distribution, the probability density function is a constant within the interval and zero outside the interval. The probability density is given by the formula:
f(x) = 1 / (b - a)
where 'a' and 'b' are the lower and upper bounds of the interval respectively (in this case, a = 73 and b = 103).
To find the 90th percentile (P90), we need to solve the following equation:
P90 = ∫(a to x) f(t) dt = 0.9
Integrating the probability density function gives:
∫(a to x) (1 / (b - a)) dt = (1 / (b - a)) * (x - a) = 0.9
Rearranging the equation, we have:
(x - a) / (b - a) = 0.9
Simplifying further:
x - a = 0.9 * (b - a)
x - 73 = 0.9 * (103 - 73)
x - 73 = 0.9 * 30
x - 73 = 27
Adding 73 to both sides of the equation, we get:
x = 73 + 27
x = 100
Therefore, the high temperature which 90% of the August days exceeded is 100 degrees Fahrenheit.
Assuming a uniform distribution:
(x-73)/(103-73) = .9
Solve for x.