Question 1192784
**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.