The number of crackers in a box of Crackerbox Crackers is normally distributed with a mean of 75 and a standard deviation of 2 crackers. What is the probability that a box has more than 76 crackers? What is the probability that a box has less than 70 crackers? What is the probability that a box has between 70 and 80 crackers?
The number of crackers in a box of Crackerbox Crackers is normally distributed with a mean of 75 and a standard deviation of 2 crackers. What is the probability that a box has more than 76 crackers? What is the probability that a box has less than 70 crackers? What is the probability that a box has between 70 and 80 crackers?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:### Understanding Cumulative Probability (CDF) for Normally Distributed Data
**Concept Overview:**
*Finding the Cumulative Probability (CDF)* allows us to understand the probability that a random variable from a given distribution will fall below a given value. Here, we are specifically dealing with normally distributed data, characterized by its mean (µ) and standard deviation (σ).
- **Mean (µ):** The average number of crackers in a box of Crackerbox Crackers is 75.
- **Standard Deviation (σ):** The standard deviation is 2 crackers, indicating the variability around the mean.
**Inverse Cumulative Distribution Function (Inverse CDF):** This mathematical function helps to find the value below which a given percentage of data falls in a normal distribution.
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**Example Problem:**
We are given a normal distribution (mean = 75, standard deviation = 2) and need to find the following probabilities:
1. **What is the probability that a box has more than 76 crackers?**
2. **What is the probability that a box has less than 70 crackers?**
3. **What is the probability that a box has between 70 and 80 crackers?**
To solve these probabilities, we can use the properties of the normal distribution and the inverse cumulative distribution function.
**Calculation Methods:**
1. **Probability of more than 76 crackers:**
- This requires finding the area under the normal distribution curve to the right of 76.
2. **Probability of less than 70 crackers:**
- This requires finding the area under the normal distribution curve to the left of 70.
3. **Probability between 70 and 80 crackers:**
- This involves finding the area under the normal distribution curve between 70 and 80.
**Graph Explanation:**
If accompanying a graph, it would typically display a bell curve (representing the normal distribution), highlight areas under the curve relating to the specified ranges (more than 76, less than 70, between 70 and 80), and shade the relevant portions.
These values can be found using statistical tools or software that handle normal distribution calculations. For an educational practice, using a Z-table or statistical calculator would be recommended.
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By understanding these concepts and methods, students can better grasp how normal distribution properties are applied in real-world scenarios, such as quality control in manufacturing processes.
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