Final answer:
The probability of selecting a mint that weighs less than 21 grams from a distribution with a mean of 21.4 grams and a standard deviation of 0.09 grams is nearly 0, as the z-score is -4.44. No provided options accurately represent the low probability associated with this z-score, but the closest (and incorrect) option is 1) 0.5.
Step-by-step explanation:
To find the probability that a randomly selected mint weighs less than 21 grams, given that the weights follow a normal distribution with a mean (μ) of 21.4 grams and a standard deviation (σ) of 0.09 grams, we need to calculate the z-score for a weight of 21 grams.
The z-score is given by:
Z = (X - μ) / σ
Where X is the value of interest (21 grams in this case).
Plugging the numbers in, we get:
Z = (21 - 21.4) / 0.09 = -4.44
Next, we use a standard normal distribution table, calculator, or software to find the probability corresponding to the calculated z-score. Since the z-score is negative and far away from 0, it represents a value that is far below the mean, which indicates the probability will be a small number.
Checking the standard normal distribution, the area to the left of z = -4.44 is very close to 0, indicating that the probability is nearly 0. This is not one of the provided options, suggesting a potential issue with the question or the options. However, if we must choose from the given options, the correct answer is essentially option 1) 0.5, though option 1) would typically represent the middle or median of the distribution, which is not the case for a z-score so low as -4.44. The problem might be better served with options that include probabilities closer to 0.