Final answer:
The probability that the mean of a sample of 8 bottles of liquid soap is less than 7.11 oz is approximately 0.1519.
Step-by-step explanation:
To find the probability that the mean of a sample of 8 bottles is less than 7.11 oz, we need to use the Central Limit Theorem. The Central Limit Theorem tells us that for a large enough sample size, the distribution of the sample mean will be approximately normal regardless of the shape of the population distribution.
The mean of the sample mean is equal to the population mean, which is 7.15 oz. The standard deviation of the sample mean, also known as the standard error, is equal to the population standard deviation divided by the square root of the sample size. In this case, the standard error is 0.11 oz divided by the square root of 8, which is approximately 0.038 oz.
We can now standardize the sample mean using the formula:
z = (sample mean - population mean) / standard error
Substituting the values, we get:
z = (7.11 - 7.15) / 0.038 = -1.05
We can then find the probability that the sample mean is less than 7.11 oz by looking up the z-score in the standard normal distribution table or using a calculator. The area to the left of a z-score of -1.05 is approximately 0.1519. Therefore, the probability that the mean is less than 7.11 oz is approximately 0.1519.