Question

The water supply board of a metropolitan city reveals that the each family consumes an average of 20 liters of drinking water per day, with a standard deviation of √ 200 liters. let x denote the drinking water consumption per family and follow the gamma distribution. 1 find α and β. 2 find the probability that a randomly selected family consumes more than 20 day.

Answer 1

To determine the parameters α and β for the gamma distribution given the mean and standard deviation, use the formulas α = (μ/σ)^2 and β = σ^2/μ. With a mean of 20 liters and a standard deviation of sqrt(200), α is approximately 2 and β is approximately 10. To find the probability of a family consuming more than 20 liters of water a day, a gamma distribution table or calculator is needed.

Step-by-step explanation:

To find the values of the parameters α (alpha) and β (beta) for the gamma distribution, given the mean and the standard deviation, we use the following formulas:

1. α = (μ/σ)^2, where μ is the mean and σ is the standard deviation.
2. β = σ^2/μ.

The mean (μ) of water consumption is given as 20 liters per day, and the standard deviation (σ) is the square root of 200, which is approximately 14.14 liters.

Now we can calculate α and β:

1. α = (20/14.14)^2 ≈ 2.
2. β = (14.14)^2/20 ≈ 10.

To find the probability that a randomly selected family consumes more than 20 liters of water a day, we would look at the gamma distribution table or calculate it using a statistical software or a gamma distribution calculator, using the found values of α and β.