Final answer:
To calculate the probability that a part is less than 19.3 cm, convert the value to a z-score and look up the area to the left of this z-score on the standard normal distribution table. For a z-score of 1.2, the probability is approximately 0.8849, or 88.49%.
Step-by-step explanation:
To find the probability that a randomly selected part is less than 19.3 centimeters when the lengths are normally distributed with a mean of 16.3 centimeters and a standard deviation of 2.5 centimeters, we use the standard normal distribution.
First, we convert the value of 19.3 centimeters into a z-score, which is the number of standard deviations away from the mean. This is done using the formula:
Z = (X - μ) / σ
Where X is the value we are looking for the probability of (19.3 cm), μ (mu) is the mean (16.3 cm), and σ (sigma) is the standard deviation (2.5 cm).
Z = (19.3 - 16.3) / 2.5 = 3 / 2.5 = 1.2
Next, we look up the z-score of 1.2 in the standard normal distribution table or use a calculator to find the area to the left of this z-score. This area gives us the probability that a randomly selected part is less than 19.3 centimeters. The exact probability will often be provided in the table or by the calculator, but for illustration, if the area to the left of the z-score is 0.8849 (for a z-score of 1.2), this is the probability we are looking for.
Therefore, the probability that the part is less than 19.3 centimeters is approximately 0.8849, or 88.49%.