A city uses a large number of rain barrels to collect water for use in maintaining green spaces around the city. After a heavy rain, the amount of water collected by each barrel is a random variable with mean 327 gallons and standard deviation 65 gallons. What is the probability that the total amount of rain water collected in 50 such barrels is more than 16,750 gallons?

Step-by-step explanation:

Hello!

The variable of interest is X: the amount of rainwater collected in a barrel (gallons)

Assuming this variable has a normal distribution with mean μ= 327 gallons and standard deviation σ= 65 gallons.

If a sample of 50 barrels is taken you need to calculate the probability of the average rainwater collected to be more than 16.750 gallons, symbolically:

P(X[bar]>16.750)

To calculate this probability you have to use the sampling distribution and the standard normal distribution.

First to transform the value of X[bar] into a value of Z (standardize the value) then you can look for the corresponding probability in the Z-table.

Using Z= (X[bar]-μ)/(σ/√n)~N(0;1)

P(X[bar]>16.750)= P(Z>(16.750-327)/(65/√50)= P(Z>-33.75)

1 - P(Z≤-33.75)= 1 - 0= 1

I hope this helps!