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A particular fruit's weights are normally distributed, with a mean of 342 grams and a standard deviation of 9 grams. If you pick 12 fruit at random, what is the probability that their mean weight will be between 337 grams and 339 grams Submit Question

A particular fruit's weights are normally distributed, with a mean of 342 grams and a standard deviation of 9 grams. If you pick 12 fruit at random, what is the probability that their mean weight will be between 337 grams and 339 grams Submit Question

A First Course in Probability (10th Edition)
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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**Title: Probability and Normal Distribution in Fruit Weights** A particular fruit's weights are normally distributed, with a mean of 342 grams and a standard deviation of 9 grams. **Problem Statement:** If you pick 12 fruit at random, what is the probability that their mean weight will be between 337 grams and 339 grams? **Instructions:** Enter your calculation in the provided text box and then click "Submit Question" to check your answer. **Guidelines for Solution:** To calculate the probability, consider the following steps: 1. **Understand the Central Limit Theorem**: Since the weights are normally distributed, the sample mean will also be normally distributed. 2. **Calculate the Standard Error of the Mean (SEM)**: - Formula: SEM = σ/√n, where σ is the standard deviation and n is the sample size. 3. **Z-Score Calculation**: - Convert the problem into a Z-score format using the formula: Z = (X - μ) / SEM, where X is the target mean, μ is the population mean, and SEM is the standard error of the mean. 4. **Find the Probability**: - Use standard normal distribution tables or software to find the probability corresponding to the calculated Z-scores. This educational task will deepen your understanding of statistical concepts such as the normal distribution and the calculation of probabilities in applied contexts.
Transcribed Image Text:**Title: Probability and Normal Distribution in Fruit Weights** A particular fruit's weights are normally distributed, with a mean of 342 grams and a standard deviation of 9 grams. **Problem Statement:** If you pick 12 fruit at random, what is the probability that their mean weight will be between 337 grams and 339 grams? **Instructions:** Enter your calculation in the provided text box and then click "Submit Question" to check your answer. **Guidelines for Solution:** To calculate the probability, consider the following steps: 1. **Understand the Central Limit Theorem**: Since the weights are normally distributed, the sample mean will also be normally distributed. 2. **Calculate the Standard Error of the Mean (SEM)**: - Formula: SEM = σ/√n, where σ is the standard deviation and n is the sample size. 3. **Z-Score Calculation**: - Convert the problem into a Z-score format using the formula: Z = (X - μ) / SEM, where X is the target mean, μ is the population mean, and SEM is the standard error of the mean. 4. **Find the Probability**: - Use standard normal distribution tables or software to find the probability corresponding to the calculated Z-scores. This educational task will deepen your understanding of statistical concepts such as the normal distribution and the calculation of probabilities in applied contexts.
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