SOLUTION: A soda machine dispenses soda into 16-ounce cups. Tests shows that the actual amount of soda dispensed has a mean of 15.85 oz and a standard deviation of 9.5 oz. A sample of 12 cup

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Question 1192784: A soda machine dispenses soda into 16-ounce cups. Tests shows that the actual amount of soda dispensed has a mean of 15.85 oz and a standard deviation of 9.5 oz. A sample of 12 cups is taken. What is the probability that the sample mean is at most 11.25?
Answer by CPhill(1959) (Show Source):
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**1. Understand the Problem**
* We are given the population mean (μ = 15.85 oz) and standard deviation (σ = 0.95 oz) of the amount of soda dispensed by a machine.
* We have a sample of 12 cups (n = 12).
* We want to find the probability that the sample mean (x̄) is at most 11.25 oz.
**2. Assumptions**
* We assume that the amount of soda dispensed in each cup is normally distributed.
* We assume that the sample is a random sample.
**3. Calculate the Standard Error of the Mean**
* The standard error of the mean (σx̄) is calculated as:
σx̄ = σ / √n
σx̄ = 0.95 / √12
σx̄ ≈ 0.2739 oz
**4. Standardize the Sample Mean**
* We need to standardize the sample mean using the z-score formula:
z = (x̄ - μ) / σx̄
z = (11.25 - 15.85) / 0.2739
z ≈ -16.76
**5. Find the Probability**
* We want to find P(x̄ ≤ 11.25), which is equivalent to finding P(z ≤ -16.76).
* Using a standard normal distribution table or a calculator, we find that P(z ≤ -16.76) is extremely close to 0.
**Conclusion:**
The probability that the sample mean of 12 cups is at most 11.25 oz is extremely low (approximately 0). This suggests that it is very unlikely to obtain such a low sample mean if the true population mean is 15.85 oz.