Math Problem Statement
Solution
To solve the problems based on the given normal distribution with and , let’s work through each part.
Part (a): Proportion between 10 and 20
For a normal distribution, we calculate probabilities using the z-score formula:
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Compute -scores for and :
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Use the standard normal distribution table or a calculator to find the cumulative probabilities:
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The proportion between and is the difference:
Part (b): Probability between 28 and 38
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Compute -scores for and :
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Use the standard normal distribution table or a calculator to find the cumulative probabilities:
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The probability between and is the difference:
Final Answers
- (a): Proportion between 10 and 20: 0.0726
- (b): Probability between 28 and 38: 0.3811
Would you like additional explanations or clarifications on how these probabilities were calculated?
Related Questions
- How do you interpret the cumulative probabilities in the standard normal distribution table?
- What are the steps for using a calculator to determine probabilities for a normal distribution?
- How would the probabilities change if the standard deviation were larger or smaller?
- How can you calculate probabilities for values outside the range provided?
- How can the z-score formula be rearranged to find given a probability?
Tip:
When working with normal distributions, always ensure your -score calculations are accurate by double-checking the formula and values substituted.
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Math Problem Analysis
Mathematical Concepts
Normal Distribution
Probability
Z-scores
Formulas
Z-score formula: z = (x - μ) / σ
Probability between two values in a normal distribution
Theorems
Properties of the Standard Normal Distribution
Suitable Grade Level
College-level Statistics or Advanced High School
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