High temperatures in a certain city for the month of August follow a uniform distribution over the interval 65°F
to 87°F. What is the probability that a randomly selected August day has a high temperature that exceeded
70°F?
i need help, what type of calculator can be used
You can play around with Z table stuff at
davidmlane.com/hyperstat/z_table.html
oops. I read normal distribution.
For a uniform distribution, every temperature has an equal chance, so you want
(87-70)/(87-65) = 17/22
To find the probability that a randomly selected August day has a high temperature that exceeds 70°F, we need to calculate the area under the probability density function (PDF) of the uniform distribution for temperatures exceeding 70°F.
Let's first determine the range of the distribution. The given interval is from 65°F to 87°F. Since it is a uniform distribution, the probability density is constant over this interval, and outside this range, the probability density is zero.
The probability density function (PDF) of a uniform distribution is given by:
f(x) = 1 / (b - a)
Where:
f(x) is the probability density
a is the lower bound of the interval (65°F)
b is the upper bound of the interval (87°F)
In this case, a = 65°F and b = 87°F. Therefore:
f(x) = 1 / (87 - 65) = 1 / 22
To find the probability of the high temperature exceeding 70°F, we need to integrate the PDF from 70°F to 87°F:
P(X > 70) = ∫[70, 87] f(x) dx
P(X > 70) = ∫[70, 87] (1 / 22) dx
P(X > 70) = [x / 22] from 70°F to 87°F
P(X > 70) = (87/22) - (70/22)
P(X > 70) = 17/22
Therefore, the probability that a randomly selected August day has a high temperature exceeding 70°F is 17/22 or approximately 0.7727 (rounded to four decimal places).