Home $> 2023$ Fall2 MAT120 (TTh11:30) > Assessment
3.3 - Normal Distributions

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Question 4
A manufacturer knows that their items have a normally distributed length, with a mean of 13.8 inches, and standard deviation of 1.8 inches.

If one item is chosen at random, what is the probability that it is less than 8.5 inches long? Round your answer to three decimal places.

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Step 1 ：First, we need to calculate the z-score, which is given by the formula \(Z = \frac{X - \mu}{\sigma}\), where \(X\) is the value we are interested in, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step 2 ：Substituting the given values into the formula, we get \(Z = \frac{8.5 - 13.8}{1.8} = -2.9444444444444446\).

Step 3 ：Next, we use the cumulative distribution function (CDF) of the normal distribution to find the probability that a random variable is less than a certain value. The CDF gives the probability that a random variable is less than a certain value.

Step 4 ：Using the calculated z-score, we find the probability to be 0.0016176751273028261.

Step 5 ：Rounding this to three decimal places, we get 0.002.

Step 6 ：Thus, the probability that a randomly chosen item is less than 8.5 inches long is \(\boxed{0.002}\).