Final answer:
a) The standard deviation of the average amount of sunscreen for a sample of 6 tubes is 2.86 ml.
b) The probability that the average amount from 6 tubes is less than 388 ml is approximately 0.26%.
Step-by-step explanation:
When dealing with the average (also called the mean) of a set of samples from a normal distribution, it's important to understand how the standard deviation of this average is calculated. Given a population standard deviation (σ) and a sample size (n), the standard deviation of the sample mean is equal to σ/√n. Applying this formula to the sunscreen tubes:
- Population standard deviation (σ) = 7 ml
- Sample size (n) = 6 tubes
To find the standard deviation of the average, σ we simply divide the population standard deviation by the square root of the sample size:
Sample mean = σ/√n
Sample mean = 7 ml/√6
The sample mean ≈ 2.86 ml
For part b, we need to determine the probability that the average of 6 tubes is less than 388 ml. To do this, we find the z-score by subtracting the population mean from the sample mean and dividing by the standard deviation of the average:
z = (X - μ) / Sample mean
= (388 ml - 396 ml) / 2.86 ml
≈ -2.8
a) The corresponding probability for a z-score of -2.8 is approximately 0.0026.
b) Therefore, there's about a 0.26% chance that the average amount from 6 tubes is less than 388 ml.