A manufacturer knows that their items have a normally distributed length, with a mean of 12.9 inches, and standard deviation of 3.1 inches.
If one item is chosen at random, what is the probability that it is less than 17.8 inches long?
To find the probability that an item chosen at random is less than 17.8 inches long, we need to find the area under the normal distribution curve to the left of 17.8 inches.
We can calculate the z-score using the formula:
z = (x - μ) / σ
where x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.
Plugging in the values, we get:
z = (17.8 - 12.9) / 3.1
z ≈ 1.5806
Now, we can use the z-score to look up the probability in the standard normal distribution table. From the table, we find that the probability corresponding to a z-score of 1.5806 is 0.9432.
Therefore, the probability that a randomly chosen item is less than 17.8 inches long is approximately 0.9432.