A bottling company uses a filling machine to fill glass bottles with mango juice. The bottles are supposed to contain 300 milliliters (ml). In fact the amounts vary according to a normal distribution with mean = 298 and standard dev = 3 ml. What is the probability that an individual bottle contains less than 295 ml?
Part 2 - What is the probability that the average content of a 4 pack of bottles is less than 295 ml?
With one bottle, you can use the Z-score.
Z = (X - mean)/SD = (295 - 298)/3 = -3/3 = -1
If you have memorized the major divisions of the areas in a normal distribution as indicated by the standard deviation (SD), you would know that 16% of the scores lie below this point = .16.
If you don't know these proportions, consult a table in the back of your statistics text called something like "areas under the normal distribution."
For the 4 bottles, you use the same formula. However, this time — since you are dealing with a sample mean rather than just an individual score — instead of dividing by the SD, you divide by the standard error of the mean (SE).
Z = (X - mean)/SE, where
SE = SD/sq.rt. of N
Solve for Z and look it up in that table to get your answer.
I hope this helps. Thanks for asking.
To find the probability that an individual bottle contains less than 295 ml, we can use the Z-score formula. The Z-score is calculated as (X - mean) / standard deviation.
In this case, X is the desired value, which is 295 ml. The mean is given as 298 ml, and the standard deviation is given as 3 ml. Plugging these values into the formula, we get:
Z = (295 - 298) / 3 = -1
Next, we need to find the corresponding area under the normal distribution curve for this Z-score. This can be done using a Z-table or statistical software. Looking up the Z-score of -1 in the table, we find that the area to the left of -1 is 0.1587.
Therefore, the probability that an individual bottle contains less than 295 ml is approximately 0.1587, or 15.87%.
Now, let's move on to part 2 of the question, which asks for the probability that the average content of a 4 pack of bottles is less than 295 ml.
To calculate this probability, we need to use the formula for the Z-score of a sample mean, which is given by (X - mean) / (standard deviation / sqrt(N)), where N is the sample size.
In this case, X is still 295 ml, the mean is 298 ml, the standard deviation is 3 ml, and N is 4 (since we have a 4 pack of bottles). Plugging these values into the formula, we get:
Z = (295 - 298) / (3 / sqrt(4)) = -2
Again, we need to find the corresponding area under the normal distribution curve for this Z-score. Looking up the Z-score of -2 in the table, we find that the area to the left of -2 is 0.0228.
Therefore, the probability that the average content of a 4 pack of bottles is less than 295 ml is approximately 0.0228, or 2.28%.