Boxes are labeled as containing 500g of cereal. The machine filling the boxes produce weights that are normally distributed with standard deviation 12g.
1) Suppose a law states that no more than 5% of a manufacturer's cereal boxes can contain less than the stated weight of 500 g. At what target weight should the manufacturer set its filling machine?
2) If (mew sign, mean symbol) is 500.8%, weigh more than y amount of grams)
I think you will be able to answer these questions if you play around a bit at
http://davidmlane.com/hyperstat/z_table.html
1) To determine the target weight the manufacturer should set its filling machine, we need to find the weight that corresponds to the 5th percentile of the normal distribution.
Step 1: Convert the 5% probability to a z-score.
Since the normal distribution is symmetric, we can use a standard normal distribution table or calculator to find the z-score that corresponds to a cumulative probability of 0.05 from the left side of the distribution.
The z-score is approximately -1.645 (you can find this using the standard normal distribution table or calculator).
Step 2: Use the z-score to find the corresponding weight value.
We can use the formula:
z = (X - mean) / standard deviation
Rearranging the formula:
X = (z * standard deviation) + mean
X = (-1.645 * 12) + 500
X ≈ 480.54
Therefore, the manufacturer should set the target weight for the filling machine to be approximately 480.54 grams to ensure that no more than 5% of the boxes contain less than 500 grams.
2) To find the weight above which a certain percentage of cereal boxes weigh, we need to find the corresponding z-score and then use it to calculate the weight.
Step 1: Convert the given percentage to a z-score.
Let's say we want to find the weight above which 80% of the cereal boxes weigh. We need to find the z-score that corresponds to a cumulative probability of 0.80 from the left side of the distribution.
The z-score is approximately 0.84 (you can find this using the standard normal distribution table or calculator).
Step 2: Use the z-score to find the corresponding weight value.
We can use the formula:
z = (X - mean) / standard deviation
Rearranging the formula:
X = (z * standard deviation) + mean
X = (0.84 * 12) + 500
X ≈ 510.08
Therefore, if the mean weight is 500 grams, more than 80% of the cereal boxes would weigh above approximately 510.08 grams.