Question 1170505
Let X be the amount of liquid filled in the pop bottles. We are given:

* Standard deviation (σ) = 25 mL
* We want 90% of the bottles to contain at least 500 mL, which means P(X ≥ 500) = 0.90.

We want to find the mean (μ) such that this condition is satisfied.

**1. Convert to Standard Normal Distribution**

We can standardize the random variable X using the formula:

Z = (X - μ) / σ

where Z follows a standard normal distribution (mean 0, standard deviation 1).

We want P(X ≥ 500) = 0.90. Let's rewrite this in terms of Z:

P((X - μ) / σ ≥ (500 - μ) / 25) = 0.90
P(Z ≥ (500 - μ) / 25) = 0.90

**2. Find the Z-score**

We need to find the z-score such that the area to the right of it is 0.90. This means the area to the left of it is 1 - 0.90 = 0.10.

Using a standard normal distribution table or a calculator, we find the z-score corresponding to a cumulative probability of 0.10. This z-score is approximately -1.28.

So, (500 - μ) / 25 = -1.28.

**3. Solve for μ**

Now, we solve for μ:

500 - μ = -1.28 * 25
500 - μ = -32
μ = 500 + 32
μ = 532

**4. Answer**

The mean should be set to 532 mL.

**Therefore, the mean should be set to 532 mL so that 90% of the bottles contain at least 500 mL.**