Math Problem Statement
Solution
This problem involves calculating the probability that a normally distributed variable is less than a certain value.
Given:
- The mean inches
- The standard deviation inches
- We need to find the probability for a normally distributed random variable .
Solution
-
Standardize the value of 10 to a z-score, which is calculated by: Substituting in the values:
-
Find the probability for using the standard normal distribution table or a calculator.
Looking up in the standard normal table or using a calculator gives approximately:
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:
- How do we interpret a z-score in a normal distribution?
- How would the probability change if the standard deviation were smaller?
- What if we wanted the probability that the item is more than 10 inches long?
- How can we use this process for values other than 10 inches?
- 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
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