Final answer:
The value at which 75% of the sample means will be less than is approximately 36.319 oz.
Step-by-step explanation:
To find the value at which 75% of the sample means will be less than, we need to calculate the z-score corresponding to the 75th percentile. The z-score is calculated using the formula: z = (x - μ) / (σ / √n), where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
In this case, the population mean (μ) is 36 oz, the population standard deviation (σ) is 0.55 oz, and the sample size (n) is 56 bottles. We want to find the value of x that corresponds to a z-score such that 75% of the sample means are less than x. This is equivalent to finding the z-score that corresponds to the 75th percentile, which we can look up in the standard normal distribution table.
Using the standard normal distribution table or a calculator, we find that the z-score corresponding to the 75th percentile is approximately 0.674. Plugging in the values, we get: 0.674 = (x - 36) / (0.55 / √56). Solving for x gives us: x = 36 + (0.674 * (0.55 / √56)). Evaluating this expression, we find that x is approximately 36.319. Therefore, 75% of the sample means will be less than 36.319 oz.