Math 14 HW 6.4.11-T Using the Central Limit Theorem

Fiorentino Siciliano
11 Mar 202311:14
EducationalLearning
32 Likes 10 Comments

TLDRThe script discusses the probability calculations for the mean weight of passengers on a water taxi. It starts by determining the maximum mean weight for a full capacity of 25 passengers, which is 150 pounds. Then, using the Central Limit Theorem, it calculates the probability that the mean weight exceeds this value when filled with 25 passengers, resulting in an extremely low probability due to a high z-score of -6.38. The script also explores a scenario with a revised capacity of 20 passengers and finds the probability of exceeding a mean weight of 187.5 pounds, which is higher at 93.45%, indicating the new capacity may not be safe.

Takeaways
  • 🚤 The water taxi's load limit is 3750 pounds, with a stated capacity for 25 passengers.
  • 📊 The passengers' weights are normally distributed with a mean of 201 pounds and a standard deviation of 40 pounds.
  • 🔢 The maximum mean weight of passengers for the taxi to be filled to capacity without exceeding the load limit is calculated to be 150 pounds.
  • 📚 According to the Central Limit Theorem, the sample means will follow a normal distribution with the given population mean and standard deviation.
  • 📉 The probability of the mean weight of 25 randomly selected passengers exceeding 150 pounds is extremely low, with a z-score of -6.38 leading to a probability close to zero.
  • 📈 The standard deviation of the sample means is calculated using the formula: population standard deviation divided by the square root of the sample size.
  • 📝 The z-score is calculated by subtracting the mean of the sample means from a given value and dividing by the standard deviation of the sample means.
  • 📉 The revised scenario with a capacity of 20 passengers and a maximum mean weight of 187.5 pounds for safety is analyzed.
  • 🔍 A z-score of -1.51 is calculated for the new scenario, indicating a higher probability of the mean weight exceeding the safe limit compared to the initial scenario.
  • 🤔 The probability of overloading with 20 passengers is over 50 percent, suggesting that the new capacity may not be safe enough.
  • 📊 The use of StatCrunch software is demonstrated to find the probabilities associated with the z-scores in the scenarios.
Q & A

  • What is the stated capacity of the water taxi in terms of the number of passengers?

    -The stated capacity of the water taxi is 25 passengers.

  • What is the maximum load limit of the water taxi in pounds?

    -The water taxi is rated for a load limit of 3750 pounds.

  • What is the mean weight of the passengers according to the normal distribution?

    -The mean weight of the passengers is 201 pounds.

  • What is the standard deviation of the passengers' weights?

    -The standard deviation of the passengers' weights is 40 pounds.

  • What is the maximum mean weight of passengers if the water taxi is filled to its stated capacity?

    -The maximum mean weight of passengers, if the water taxi is filled to its stated capacity of 25 passengers, is 150 pounds.

  • According to the Central Limit Theorem, what is the distribution of the sample means?

    -According to the Central Limit Theorem, the sample means will have a normal distribution with the mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size (N).

  • What is the probability that the mean weight of 25 randomly selected passengers exceeds 150 pounds?

    -The probability that the mean weight of 25 randomly selected passengers exceeds 150 pounds is extremely low, as indicated by a z-score calculation resulting in a probability of 0.000.

  • What is the z-score for a sample mean of 150 pounds with a population mean of 201 and a standard deviation of 40 pounds for a sample size of 25?

    -The z-score for this scenario is approximately -6.38, indicating a value far below the mean of the sample means distribution.

  • If the water taxi's capacity is revised to 20 passengers, what is the new maximum mean weight that does not cause the total load to exceed 3750 pounds?

    -With a revised capacity of 20 passengers, the new maximum mean weight that does not cause the total load to exceed 3750 pounds is 187.5 pounds.

  • What is the probability that the mean weight of 20 randomly selected passengers exceeds 187.5 pounds?

    -The probability that the mean weight of 20 randomly selected passengers exceeds 187.5 pounds is approximately 93.45%, which is calculated using a z-score of -1.51.

  • Is the new capacity of 20 passengers considered safe based on the probability of overloading?

    -No, the new capacity of 20 passengers is not considered safe, as the probability of overloading is over 50%, indicating a high risk of exceeding the load limit.

Outlines
00:00
🚤 Maximum Mean Weight Calculation for a Water Taxi

This paragraph discusses a scenario where a water taxi is carrying passengers with weights normally distributed around a mean of 201 pounds and a standard deviation of 40 pounds. The taxi has a capacity for 25 passengers and a load limit of 3750 pounds. The script calculates the maximum mean weight of passengers the taxi can carry without exceeding its load limit, which is found to be 150 pounds. It then explores the probability that the mean weight of 25 randomly selected passengers exceeds this maximum mean weight, using the central limit theorem to determine the sample mean's distribution. The z-score calculation for this scenario results in a value of -6.38, indicating an extremely low probability of the mean weight exceeding 150 pounds.

05:01
📊 Probability of Exceeding Mean Weight with Revised Capacity

The second paragraph delves into a hypothetical situation where the water taxi's capacity is revised to 20 passengers. The script calculates the new maximum mean weight that would not cause the total load to exceed 3750 pounds, which is 187.5 pounds. It then finds the z-score for this new scenario, resulting in -1.51, and uses it to determine the probability that the mean weight of 20 randomly selected passengers exceeds 187.5 pounds. The probability is found to be 0.9345, or 93.45%, indicating a high likelihood of the water taxi being overloaded if the mean weight of passengers is above this new maximum mean weight.

10:02
🚫 Assessing the Safety of the Revised Passenger Capacity

In the final paragraph, the script evaluates the safety of the revised capacity of 20 passengers. Given the high probability of overloading calculated in the previous paragraph, the script concludes that the new capacity does not appear to be safe enough, as there is a greater than 50% chance of the water taxi exceeding its load limit with a mean weight of passengers above the maximum allowed mean weight.

Mindmap
Keywords
💡Water Taxi
A water taxi is a type of waterborne transportation service that carries passengers from one location to another, typically across a body of water. In the video's context, it is the main subject of discussion as it pertains to the capacity and weight limits for passengers. The script discusses the water taxi's capacity and how it is rated for a load limit, which is crucial for understanding the safety and operational aspects of the service.
💡Normal Distribution
Normal distribution, also known as Gaussian distribution, is a probability distribution that is characterized by a symmetrical bell-shaped curve. In the script, it is mentioned that the weights of the passengers are normally distributed with a mean of 201 pounds and a standard deviation of 40 pounds. This statistical concept is central to the video as it helps in calculating probabilities related to the passengers' weights and the water taxi's capacity.
💡Mean Weight
Mean weight refers to the average weight of a group of items or individuals. In the video, the mean weight of the passengers is given as 201 pounds, which is used to calculate the maximum mean weight that the water taxi can carry without exceeding its load limit. The concept is essential for understanding the statistical analysis performed in the script.
💡Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the script, the standard deviation of the passengers' weights is 40 pounds, indicating the spread of individual weights around the mean. It plays a significant role in the video as it is used to determine the variability of the passengers' weights and to calculate the z-scores for probability assessments.
💡Capacity
Capacity, in the context of the video, refers to the maximum number of passengers that the water taxi can safely carry. The stated capacity is 25 passengers. This term is key to understanding the operational limits of the water taxi and how it relates to the safety and efficiency of the service.
💡Load Limit
Load limit is the maximum weight that a vehicle or vessel can safely carry. The water taxi is rated for a load limit of 3750 pounds. This term is crucial as it sets the threshold for the total weight of passengers and cargo that the water taxi can handle without compromising safety.
💡Central Limit Theorem
The Central Limit Theorem is a statistical theory that states that the distribution of sample means will approach a normal distribution as the sample size becomes larger, regardless of the shape of the population distribution. In the script, it is mentioned in relation to the sample means of the passengers' weights, which will have a normal distribution with a specific mean and standard deviation, allowing for the calculation of probabilities.
💡Sample Size
Sample size refers to the number of observations or individuals in a sample. In the video, the sample size is initially 25 passengers when the water taxi is filled to its stated capacity. Later, the sample size is revised to 20 passengers in a scenario. The sample size is important for calculating the standard deviation of the sample means and for determining the probabilities in the statistical analysis.
💡Z-Score
A z-score is a measure of how many standard deviations an element is from the mean of a set of numbers. In the script, z-scores are calculated to determine the probabilities of the mean weight of passengers exceeding certain values. The z-score is used in the context of the normal distribution to find the area under the curve, which corresponds to the probability.
💡Probability
Probability is a measure of the likelihood that a given event will occur. In the video, probabilities are calculated to assess the risk of the water taxi exceeding its load limit when filled with passengers. The script discusses how to find these probabilities using z-scores and the normal distribution, which is essential for understanding the safety analysis of the water taxi's operation.
💡Overloading
Overloading refers to the situation where the total weight of passengers or cargo exceeds the load limit of a vehicle or vessel. In the script, the concept of overloading is discussed in the context of the water taxi's capacity and the calculated probabilities of exceeding the mean weight limit. It is a critical term for understanding the safety implications of the statistical analysis presented in the video.
Highlights

A water taxi's capacity and load limit are analyzed with a statistical approach.

Passenger weights are assumed to be normally distributed with a mean of 201 pounds and a standard deviation of 40 pounds.

The water taxi's stated capacity is 25 passengers and its load limit is 3750 pounds.

Calculation of the maximum mean weight of passengers if the water taxi is filled to its stated capacity.

Determination that the maximum mean weight is 150 pounds for the water taxi to operate within its load limit.

Application of the Central Limit Theorem to determine the distribution of sample means.

The sample mean's distribution has a mean equal to the population mean and a standard deviation dependent on sample size.

Graphical representation of the normal distribution to visualize the problem.

Calculation of the probability that the mean weight of 25 randomly selected passengers exceeds 150 pounds.

Use of the z-score formula to standardize the distribution and calculate probabilities.

A z-score of negative 6.38 indicates an extremely low probability of the mean weight exceeding 150 pounds.

Utilization of statistical software to find the probability associated with the z-score.

A probability of 1.000 suggests that it is virtually impossible for the mean weight to exceed 150 pounds with 25 passengers.

Revised assumptions for a new capacity of 20 passengers and the calculation of the corresponding maximum mean weight.

The new maximum mean weight is determined to be 187.5 pounds for a 20-passenger capacity.

Recalculation of the z-score for the new scenario with 20 passengers and a mean weight of 187.5 pounds.

A z-score of negative 1.51 for the new scenario indicates a significantly higher probability of exceeding the mean weight.

Statistical software is used again to find the probability of the new z-score, resulting in 0.9345.

Interpretation of the results suggests that the new 20-passenger capacity may not be safe due to a high probability of overloading.

Transcripts

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Related Tags
Water TaxiProbabilityNormal DistributionLoad LimitPassenger WeightStatistical AnalysisCentral Limit TheoremSample MeanZ-ScoreSafety Assessment