Question

The full amount of soft drink bottles is normally distributed, with a mean of 2.0 liters and a standard deviation of 0.05 liter. If you select a random sample of 25 bottles, what is the probability that the sample mean will be.

a. between 1.99 and 2.0 liters?

Answer 1

To find the probability that the sample mean will be between 1.99 and 2.0 liters, calculate the z-scores for both values and find the area under the normal distribution curve between those z-scores. The probability is 0.1554, or 15.54%.

Step-by-step explanation:

In order to find the probability that the sample mean will be between 1.99 and 2.0 liters, we need to calculate the z-scores for both values and find the area under the normal distribution curve between those z-scores.

First, we calculate the z-score for 1.99 liters:

z = (x - µ) / σ

z = (1.99 - 2.0) / 0.05 = -0.02 / 0.05 = -0.4

Next, we calculate the z-score for 2.0 liters:

z = (x - µ) / σ

z = (2.0 - 2.0) / 0.05 = 0 / 0.05 = 0

Using a standard normal distribution table or a calculator, we find the corresponding probability for each z-score:

For z = -0.4, the corresponding probability is 0.3446.

For z = 0, the corresponding probability is 0.5.

To find the probability between -0.4 and 0, we subtract the probability for z = -0.4 from the probability for z = 0:

0.5 - 0.3446 = 0.1554

So, the probability that the sample mean will be between 1.99 and 2.0 liters is 0.1554, or 15.54%.