Assume that the weight loss for the first two months of a diet program has a uniform distribution over the interval 6 to 12 pounds. Find the probability that a person on this diet
loses more than 11 pounds in the first two months.
To find the probability that a person on this diet loses more than 11 pounds in the first two months, we need to determine the area under the probability density function (PDF) curve for the uniform distribution.
First, let's define the variables:
- W: Weight loss in pounds
Given:
- The weight loss for the first two months follows a uniform distribution over the interval 6 to 12 pounds.
To find the probability, we need to calculate the area under the PDF curve for the range of weight loss greater than 11 pounds.
The PDF for a uniform distribution is given by:
f(w) = 1 / (b - a), for a <= w <= b
where a and b are the lower and upper bounds of the interval, respectively.
In this case, a = 6 and b = 12.
To calculate the probability that a person on this diet loses more than 11 pounds, we need to find the area under the PDF curve for w > 11.
Since the PDF for a uniform distribution is a constant within the defined range, the probability can be calculated as:
P(W > 11) = (Area of the range of weight loss > 11) / (Total area under the PDF curve)
To find the area of the range of weight loss greater than 11, we subtract the area of the range 6 to 11 from the total area under the PDF curve.
Area of the range of weight loss > 11 = (b - 11) / (b - a)
Therefore, the probability that a person on this diet loses more than 11 pounds in the first two months is:
P(W > 11) = (12 - 11) / (12 - 6) = 1/6 = 0.1667
So, the probability that a person on this diet loses more than 11 pounds in the first two months is approximately 0.1667 or 16.67%.