A machine is used to fill soda bottles in a factory. The bottles are labeled as containing 2.0 liters, but extra room at the top of the bottle allows for a maximum of 2.25 liters of soda before the bottle overflows. The standard deviation of the amount of soda put into the bottles by the machine is known to be .15 liters.

if the machine is set to fill the bottles with an average of 2.08 liters, what proportion of bottles will be overfilled?

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To determine the proportion of bottles that will be overfilled, we need to calculate the z-score for the maximum allowable amount.

The z-score is calculated using the formula: z = (x - μ) / σ

Where:
x = maximum allowable amount (2.25 liters)
μ = average amount (2.08 liters)
σ = standard deviation (0.15 liters)

Substituting the values into the formula, we get:
z = (2.25 - 2.08) / 0.15 = 1.13

Next, we need to find the proportion of bottles that have a z-score greater than 1.13. We can use a standard normal distribution table or a calculator to find this proportion.

Looking up the z-score of 1.13 in the standard normal distribution table, we find that the proportion of bottles filled above 2.25 liters is approximately 0.1309.

Therefore, approximately 13.09% of bottles will be overfilled.

To find the proportion of bottles that will be overfilled, you need to determine the probability of a bottle being filled with more than 2.25 liters of soda.

First, let's calculate the z-score for 2.25 liters using the formula:

z = (x - μ) / σ

Where:
x = 2.25 (the value we want to calculate the z-score for)
μ = 2.08 (mean value)
σ = 0.15 (standard deviation)

z = (2.25 - 2.08) / 0.15
z = 0.17 / 0.15
z = 1.13

Next, we need to find the area under the standard normal curve to the right of z = 1.13. This represents the proportion of bottles that will be overfilled.

Using a standard normal distribution table or a calculator, we find that the area to the right of z = 1.13 is approximately 0.1292.

Therefore, approximately 12.92% (0.1292 as a decimal) of the bottles will be overfilled.