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?
Think about the idea behind sampling and that of the importance of standard error.
Use z-scores and the z-score formula.
Formula: z = (x - mean)/sd -->sd = standard deviation
Your values to plug into the formula are:
x = 295
mean = 298
sd = 3
Once you determine the z-score, check a z-table (normal distribution table) for the probability. Remember that the question is asking "less than 295 ml" so you will need to take that into consideration when looking at the table.
It helps to know how to read tables to do these kinds of problems.
For part 2, you will need to take the sample size into consideration by using this formula: z = (x - mean)/(sd/√n)
...then determine your probability using the table again.
I hope this will help get you started.
To find the probability that an individual bottle contains less than 295 ml, we need to calculate the z-score using the formula:
z = (x - mean) / sd
where x is 295, mean is 298, and sd is 3.
Let's plug in the values:
z = (295 - 298) / 3
z = -1 / 3 = -0.33
Now, we can check the z-table (normal distribution table) to find the probability associated with the z-score of -0.33.
Looking up the z-score -0.33 in the table, we find that the probability corresponding to this z-score is approximately 0.3707.
So, the probability that an individual bottle contains less than 295 ml is approximately 0.3707, or 37.07%.
Now, let's move on to part 2.
To calculate the probability that the average content of a 4-pack of bottles is less than 295 ml, we need to consider the sample size and use the formula:
z = (x - mean) / (sd / sqrt(n))
Here, x and mean remain the same (295 and 298, respectively), sd is still 3, and n is the sample size, which is 4.
Plugging in the values:
z = (295 - 298) / (3 / sqrt(4))
z = -3 / (3 / 2) = -2
Now, we can consult the z-table again with a z-score of -2 to find the corresponding probability.
Finding the z-score -2 in the table, we see that the probability associated with it is approximately 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%.