Math Problem Statement

A manufacturer knows that their items have a normally distributed length, with a mean of 12.4 inches, and standard deviation of 4.1 inches. If one item is chosen at random, what is the probability that it is less than 10 inches long?

Solution

This problem involves calculating the probability that a normally distributed variable is less than a certain value.

Given:

  • The mean μ=12.4\mu = 12.4 inches
  • The standard deviation σ=4.1\sigma = 4.1 inches
  • We need to find the probability P(X<10)P(X < 10) for a normally distributed random variable XX.

Solution

  1. Standardize the value of 10 to a z-score, which is calculated by: Z=XμσZ = \frac{X - \mu}{\sigma} Substituting in the values: Z=1012.44.1=2.44.10.585Z = \frac{10 - 12.4}{4.1} = \frac{-2.4}{4.1} \approx -0.585

  2. Find the probability for Z<0.585Z < -0.585 using the standard normal distribution table or a calculator.

    Looking up Z=0.585Z = -0.585 in the standard normal table or using a calculator gives approximately: P(Z<0.585)0.2794P(Z < -0.585) \approx 0.2794

Conclusion

The probability that a randomly selected item is less than 10 inches long is approximately 0.2794 or 27.94%.

Would you like further explanation on this process or have any questions?


Here are some related questions that might deepen your understanding:

  1. How do we interpret a z-score in a normal distribution?
  2. How would the probability change if the standard deviation were smaller?
  3. What if we wanted the probability that the item is more than 10 inches long?
  4. How can we use this process for values other than 10 inches?
  5. How does a z-score help in comparing different data points within a normal distribution?

Tip: Always check if the values provided require a one-tailed or two-tailed probability, as it affects the interpretation and calculation of results.

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Math Problem Analysis

Mathematical Concepts

Probability
Normal Distribution
Z-scores

Formulas

Z = (X - μ) / σ

Theorems

Properties of the Normal Distribution

Suitable Grade Level

Grades 10-12