A company that manufactures and bottles apple juice has a machine that automatically fills 16-ounce bottles. There is some variation, however, in the exact amount of juice dispensed into each bottle. From a large number of observations taken over a long period of time, it was found that the actual amount of juice dispensed into each bottle was normally distributed with a mean of 16 ounces and a standard deviation of 1 ounce. Find the percentage of all bottles that are either underfilled or overfilled by at least 0.25 ounce.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score. Remember to calculate for both directions.
38.30%
To find the percentage of bottles that are either underfilled or overfilled by at least 0.25 ounce, we need to calculate the area under the normal distribution curve outside of the range of 15.75 to 16.25 ounces.
Step 1: Standardize the range
To do this, we need to convert the range of 15.75 to 16.25 ounces into z-scores.
Z-score formula: z = (x - μ) / σ
Where:
x = the value we want to standardize
μ = mean of the distribution
σ = standard deviation of the distribution
Calculating the z-scores for 15.75 and 16.25:
z1 = (15.75 - 16) / 1 = -0.25
z2 = (16.25 - 16) / 1 = 0.25
Step 2: Calculate the area under the curve
To find the area under the normal distribution curve outside the range of z1 and z2, we can subtract the area under the curve between z1 and z2 from 1.
P(z < z1) = P(z > -0.25)
Using a standard normal distribution table or calculator, we find that P(z > -0.25) = 0.59871.
P(z < z2) = P(z < 0.25)
Similarly, we find that P(z < 0.25) = 0.59871.
Therefore, the area under the curve between z1 and z2 is 0.59871 - 0.59871 = 0.
Step 3: Calculate the percentage
Finally, we can calculate the percentage of bottles that are either underfilled or overfilled by at least 0.25 ounce:
Percentage = Area under the curve outside the range * 100 = (1 - 0) * 100 = 100%.
Thus, the percentage of bottles that are either underfilled or overfilled by at least 0.25 ounce is 100%.