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A magazine reported that on average, American drivers will be involved in 3.4 car accidents while driving over their lifetime, with a standard deviation of 0.68 Number of accidents for American drivers follows a normal distribution. Four Americans who are no longer driving were randomly selected. What is the probability that their average number of accidents while driving will be larger than 47 P(Z > 125) For this problem, we were told that the distribution is normal. If the shape of the distribution was not stated, we would not have been able to use the normal distribution table to find the probability. What does the Central Limit Theorem tell us to do to be able to normalize the distribution? What is the rule of thumb for normalizing the sampling distribution of means?

A magazine reported that on average, American drivers will be involved in 3.4 car accidents while driving over their lifetime, with a standard deviation of 0.68 Number of accidents for American drivers follows a normal distribution. Four Americans who are no longer driving were randomly selected. What is the probability that their average number of accidents while driving will be larger than 47 P(Z > 125) For this problem, we were told that the distribution is normal. If the shape of the distribution was not stated, we would not have been able to use the normal distribution table to find the probability. What does the Central Limit Theorem tell us to do to be able to normalize the distribution? What is the rule of thumb for normalizing the sampling distribution of means?

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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**Educational Transcription: Sampling Distribution and Normalization in Statistics** --- ### Problem Overview A magazine reported that, on average, American drivers will be involved in 3.4 car accidents during their lifetime, with a standard deviation of 0.68. The number of accidents for American drivers follows a normal distribution. Four Americans who are no longer driving were randomly selected. **Problem Question**: What is the probability that their average number of accidents while driving will be larger than 4? The question can be represented as: \[ P(\overline{X} > 125) = \; \_\_\_\_ \] --- ### Explanation and Application of the Central Limit Theorem For this problem, it was stated that the distribution is normal. Normally, if the shape of the distribution is not known, the normal distribution table cannot be used to find the probability. **Central Limit Theorem Insight**: The Central Limit Theorem helps in normalizing the distribution: --- ### Method for Normalizing the Sampling Distribution In this context, one of the actions suggested for normalization can be selected from: - Increase the standard deviation - Increase the mean - Increase the sample size - It cannot be normalized --- ### Rule of Thumb for Normalizing the Sampling Distribution of Means To determine when a sample size is large enough to assume that the sampling distribution of the mean is approximately normal, one might consider these options: - \( np \geq 10 \) and \( nq \geq 10 \) - \( n > 30 \) - \( np \geq 30 \) and \( nq \geq 30 \) - \( n > 10 \) These criteria help decide appropriate sample sizes for applying the Central Limit Theorem reliably. --- This educational discussion is aimed at aiding the understanding of probability and distribution in practical statistical problems in relation to real-world scenarios.
Transcribed Image Text:**Educational Transcription: Sampling Distribution and Normalization in Statistics** --- ### Problem Overview A magazine reported that, on average, American drivers will be involved in 3.4 car accidents during their lifetime, with a standard deviation of 0.68. The number of accidents for American drivers follows a normal distribution. Four Americans who are no longer driving were randomly selected. **Problem Question**: What is the probability that their average number of accidents while driving will be larger than 4? The question can be represented as: \[ P(\overline{X} > 125) = \; \_\_\_\_ \] --- ### Explanation and Application of the Central Limit Theorem For this problem, it was stated that the distribution is normal. Normally, if the shape of the distribution is not known, the normal distribution table cannot be used to find the probability. **Central Limit Theorem Insight**: The Central Limit Theorem helps in normalizing the distribution: --- ### Method for Normalizing the Sampling Distribution In this context, one of the actions suggested for normalization can be selected from: - Increase the standard deviation - Increase the mean - Increase the sample size - It cannot be normalized --- ### Rule of Thumb for Normalizing the Sampling Distribution of Means To determine when a sample size is large enough to assume that the sampling distribution of the mean is approximately normal, one might consider these options: - \( np \geq 10 \) and \( nq \geq 10 \) - \( n > 30 \) - \( np \geq 30 \) and \( nq \geq 30 \) - \( n > 10 \) These criteria help decide appropriate sample sizes for applying the Central Limit Theorem reliably. --- This educational discussion is aimed at aiding the understanding of probability and distribution in practical statistical problems in relation to real-world scenarios.
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