Question

Assume the fill amount of bottles of a soft drink is normally distributed with a mean of 2.0 liters and standard deviation of 0.04 liter. If we take random samples of 12 bottles, find the sample mean fill amount that is lower than 95% of all such sample means.

Answer 1

To find the sample mean fill amount that is lower than 95% of all such sample means, we can use the concept of the sampling distribution of the sample mean.

In this case, the population mean (μ) is 2.0 liters, and the standard deviation (σ) is 0.04 liter. The sample size (n) is 12.

The standard deviation of the sampling distribution of the sample mean, also known as the standard error (SE), can be calculated using the formula:

SE = σ / sqrt(n)

SE = 0.04 / sqrt(12)
SE ≈ 0.01155

Now, we can calculate the z-score corresponding to the lower 95% confidence level. For a 95% confidence level, the z-score is approximately -1.645.

Next, we can find the corresponding value of the sample mean using the formula:

Sample Mean = μ + (z * SE)

Sample Mean = 2.0 + (-1.645 * 0.01155)
Sample Mean ≈ 1.981

Therefore, the sample mean fill amount that is lower than 95% of all such sample means is approximately 1.981 liters.