Question

The amount of liquid in a particular group of bottles is normally distributed. The mean is 1.95 L and the standard deviation is 0.15L. If a bottle is chosen at random, find the probability that the amount of water is between 1.88 L and 2.15 L. Show all work

Answer 1

0.5814

Explanation:

We can use the standard normal distribution to solve this problem. First, we can standardize the random variable X using the formula:

Z = (X - μ) / σ

where X is the amount of liquid in a bottle, μ is the mean of the distribution, and σ is the standard deviation of the distribution. Substituting the given values, we get:

Z = (X - 1.95) / 0.15

Now, we want to find the probability that the amount of water is between 1.88 L and 2.15 L, i.e., P(1.88 ≤ X ≤ 2.15). This is equivalent to finding the probability that the standardized random variable Z is between:

Z1 = (1.88 - 1.95) / 0.15 = -0.467

and

Z2 = (2.15 - 1.95) / 0.15 = 1

Using a standard normal table or a calculator, we can find the area under the standard normal curve between Z1 and Z2 to get the probability:

P(-0.467 ≤ Z ≤ 1) = 0.5814

Now the probability that the amount of water is between 1.88 L and 2.15 L is approximately 0.5814.