Final answer:
To find the probability of an item being less than 8.4 inches long in a normally distributed set with mean of 10.3 and standard deviation of 1.5, calculate the z-score and then find the corresponding probability from the z-table or a calculator. The probability is approximately 0.1020, or 10.20%.
Step-by-step explanation:
The student's question pertains to finding the probability that a randomly chosen item from a normally distributed set of items has a length less than a certain value. The steps to find this probability are as follows:
- Identify the mean (μ) and standard deviation (σ) of the normal distribution. In this case, μ = 10.3 inches and σ = 1.5 inches.
- Calculate the z-score of the desired measurement (8.4 inches) using the formula: z = (X - μ) / σ, where X is the value in question. Substituting the known values gives us z = (8.4 - 10.3) / 1.5.
- Compute the z-score, which is z = -1.27 (rounded to two decimal places).
- Using a standard normal distribution table, z-table, or a calculator with a normal distribution function, find the probability corresponding to the calculated z-score.
- The probability (P) of an item being less than 8.4 inches is then P(Z < -1.27).
- Looking up this z-score on the z-table or using a calculator, the probability is approximately 0.1020.
Therefore, the probability that one item is less than 8.4 inches long is 0.1020, or 10.20%.