To find the probabilities in the questions, we will use the standard normal distribution with a mean of 20.1 oz and a standard deviation of 0.4 oz.
1. Probability of a randomly sampled can containing 19.2 oz or less of soup:
We need to find the area under the normal distribution curve to the left of 19.2 oz. We can calculate the z-score for 19.2 oz as follows:
z = (19.2 - 20.1) / 0.4 = -2.25
Using a standard normal distribution table or a calculator, we can find the probability corresponding to the z-score of -2.25. The probability is approximately 0.0122 or 1.22%.
Therefore, the probability that a randomly sampled can of soup will contain 19.2 oz or less is approximately 0.0122 or 1.22%.
2. Probability of a randomly sampled can containing between 19 oz and 21 oz of soup:
We need to find the area under the normal distribution curve between 19 oz and 21 oz. We can calculate the z-scores for both values as follows:
z1 = (19 - 20.1) / 0.4 = -2.75
z2 = (21 - 20.1) / 0.4 = 2.25
Using a standard normal distribution table or a calculator, we can find the probabilities corresponding to the z-scores of -2.75 and 2.25. The probability for -2.75 is approximately 0.00339 or 0.339%, and the probability for 2.25 is approximately 0.9878 or 98.78%.
To find the probability between 19 oz and 21 oz, we subtract the probability of 19 oz or less from the probability of 21 oz or less:
0.9878 - 0.00339 ≈ 0.98441 or 98.441%
Therefore, the probability that a randomly sampled can of soup will contain between 19 oz and 21 oz is approximately 0.98441 or 98.441%.
3. Smallest (cut-off) volume of soup that 90% of the cans will contain (or exceed):
We need to find the z-score corresponding to the 90th percentile (or the area to the left of the cut-off volume) of the normal distribution. We can use a standard normal distribution table or a calculator to find the z-score.
The z-score corresponding to the 90th percentile is approximately 1.282.
Now, we can calculate the actual volume of soup using the z-score:
x = z * standard deviation + mean
x = 1.282 * 0.4 + 20.1
x ≈ 20.6
Therefore, the smallest (cut-off) volume of soup that 90% of the cans will contain (or exceed) is approximately 20.6 oz.