Sampling distribution example problem | Probability and Statistics | Khan Academy

Khan Academy
27 Oct 201014:28
EducationalLearning
32 Likes 10 Comments

TLDRThe script explores the probability of running out of water during a nature trip with 50 men, given an average water consumption of 2 liters per person with a standard deviation of 0.7 liters. It explains the concept of the sampling distribution of the sample mean and how it relates to the normal distribution. The presenter calculates the standard deviation of the sample mean and uses a Z-table to find the probability of the sample mean exceeding 2.2 liters, concluding with a 2.17% chance of water shortage.

Takeaways
  • 💧 The average male consumes 2 liters of water when active outdoors with a standard deviation of 0.7 liters.
  • 📈 A normal distribution is assumed for the water consumption of the average man, even though the actual distribution is not confirmed.
  • 🧭 Planning involves a full-day nature trip for 50 men with 110 liters of water, aiming to estimate the probability of running out.
  • 📦 The critical threshold for water shortage is set at an average consumption of 2.2 liters per man for the group.
  • 🔢 The mean water consumption per man is calculated by dividing the total amount of water (110 liters) by the number of men (50).
  • 📊 The sampling distribution of the sample mean is considered to be normally distributed due to the Central Limit Theorem.
  • 📉 The standard deviation of the sampling distribution is calculated as the population standard deviation (0.7 liters) divided by the square root of the sample size (50), resulting in 0.099 liters.
  • 🌐 The probability of running out of water is equivalent to the probability that the sample mean exceeds 2.2 liters per man.
  • 📝 To find this probability, calculate the Z-score, which is the difference between the critical value (2.2 liters) and the mean (2 liters) divided by the standard deviation of the sampling distribution (0.099 liters).
  • 🔑 Using a Z-table, the probability of the sample mean being more than 2.02 standard deviations above the mean is found to be 0.9783.
  • 🎯 The final step is to subtract this probability from 1 to find the chance of running out of water, which is approximately 2.17%.
Q & A

  • What is the average amount of water a male drinks when active outdoors according to the script?

    -The average male drinks 2 liters of water when active outdoors.

  • What is the standard deviation of the water consumption in this scenario?

    -The standard deviation of the water consumption is 0.7 liters.

  • How many liters of water are being planned to be brought for a nature trip for 50 men?

    -110 liters of water are planned to be brought for the nature trip for 50 men.

  • What is the probability calculation about in the context of the script?

    -The probability calculation is about the chance of running out of water during the nature trip.

  • What is the average water consumption per man if there are 110 liters for 50 men?

    -If there are 110 liters for 50 men, the average water consumption per man would be 2.2 liters.

  • Why is the sampling distribution of the sample mean considered to be a normal distribution?

    -The sampling distribution of the sample mean is considered to be a normal distribution due to the Central Limit Theorem, which states that the distribution of sample means will approach a normal distribution as the sample size increases, regardless of the shape of the population distribution.

  • How is the standard deviation of the sampling distribution of the sample mean calculated?

    -The standard deviation of the sampling distribution of the sample mean is calculated by dividing the population standard deviation by the square root of the sample size (n).

  • What is the Z-score and why is it important in this context?

    -The Z-score is a measure of how many standard deviations an element is from the mean. It is important in this context because it helps determine the probability of the sample mean being more than 2.2 liters per man, which would indicate running out of water.

  • What does the Z-table represent and how is it used in this scenario?

    -The Z-table represents the cumulative probability of a standard normal distribution. It is used in this scenario to find the probability of the sample mean being more than 2.02 standard deviations above the mean, which corresponds to a Z-score of 2.02.

  • What is the final probability calculated for running out of water during the nature trip?

    -The final probability calculated for running out of water during the nature trip is 2.17%.

  • How does the script suggest that the distribution of water consumption might look like for the average man?

    -The script suggests that the distribution might be skewed, with most men needing around the average of 2 liters, some needing less, and others needing more, but no one needing more than about 4 liters.

Outlines
00:00
💧 Calculating Water Needs for a Nature Trip

This paragraph discusses the statistical approach to determining the probability of running out of water during a nature trip with 50 men. The average water consumption is given as 2 liters with a standard deviation of 0.7 liters. The trip organizer plans to bring 110 liters of water and seeks to understand the likelihood of this amount being insufficient. The explanation involves understanding the distribution of water needs, calculating the sample mean, and determining the probability that the average consumption per man will exceed 2.2 liters, which would deplete the water supply.

05:08
📊 Understanding the Sampling Distribution of the Sample Mean

The second paragraph delves into the concept of the sampling distribution of the sample mean, explaining how it applies to the water consumption scenario. It clarifies that even if the original distribution of water needs is not normal, the sampling distribution of the sample mean will be normal due to the Central Limit Theorem. The mean of this distribution is the same as the population mean, 2 liters, but the standard deviation is reduced by a factor of the square root of the sample size, resulting in a standard deviation of 0.099 liters for the sample mean. The goal is to find the probability that the sample mean exceeds 2.2 liters, which would indicate a water shortage.

10:09
📉 Using the Z-Score to Determine Water Shortage Probability

The final paragraph concludes the analysis by calculating the Z-score to find the probability of the sample mean exceeding 2.2 liters. The Z-score is calculated as the difference between the critical value (2.2 liters) and the mean (2 liters), divided by the standard deviation of the sampling distribution (0.099 liters), resulting in a Z-score of 2.02. By consulting a Z-table, the probability of being more than 2.02 standard deviations above the mean is found to be 0.9783. Subtracting this from 1 gives the probability of running out of water, which is 0.0217, or 2.17%. This step-by-step process provides a clear statistical method to assess the risk of water shortage for the planned trip.

Mindmap
Keywords
💡Average
The term 'average' in the context of the video refers to the mean value of a set of data, which is calculated by summing all the values and then dividing by the number of values. It is central to the video's theme as it is used to describe the typical amount of water a male consumes when active outdoors, which is 2 liters. The average is essential for understanding the probability calculations related to the water consumption of a group of 50 men on a nature trip.
💡Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the video, it is used to quantify the spread of water consumption around the average. With a standard deviation of 0.7 liters, it indicates that the majority of males will consume an amount of water that is close to the average of 2 liters, but some may consume significantly more or less. This concept is crucial for determining the probability of running out of water on the trip.
💡Probability
Probability is the measure of the likelihood that a particular event will occur. In the video, the concept is used to calculate the chance that the group of 50 men will run out of water during their nature trip. The script discusses how to determine this probability by analyzing the distribution of water consumption and using statistical methods.
💡Liters
The term 'liters' is a unit of volume used to measure the amount of water. In the video, it is the unit used to quantify both the average water consumption per person and the total amount of water brought for the trip. The script uses liters to establish the quantities involved in the calculations and to determine if there is enough water for the trip.
💡Nature Trip
A 'nature trip' is an outdoor recreational activity, often involving physical activity in a natural setting. The video revolves around planning for such a trip, specifically addressing the logistical aspect of providing sufficient water for 50 men participating in the trip. The nature of the trip is important as it influences the water consumption of the participants.
💡Sample Mean
The 'sample mean' is the average of a subset of data drawn from a larger population. In the video, the sample mean is used to represent the average water consumption of the 50 men on the nature trip. The script discusses the probability that this sample mean will exceed a certain value, which would indicate running out of water.
💡Sampling Distribution
The 'sampling distribution' of a statistic is the probability distribution of that statistic when all possible samples of a given size are taken from the population. In the video, the sampling distribution of the sample mean is discussed to understand the likelihood of the sample mean exceeding 2.2 liters per person, which would lead to running out of water.
💡Normal Distribution
A 'normal distribution' is a type of continuous probability distribution that is characterized by a symmetrical bell-shaped curve. The video explains that even if the original distribution of water consumption is not normal, the sampling distribution of the sample mean will be approximately normal due to the Central Limit Theorem, which is key to the probability calculations.
💡Z-Score
A 'Z-score' represents the number of standard deviations an element is from the mean of a set of numbers. In the video, the Z-score is used to determine how many standard deviations above the mean the sample mean would need to be to result in running out of water. The script calculates the Z-score to find the probability associated with this event.
💡Z-Table
A 'Z-table' is a statistical table used to look up the probability that a random variable from a standard normal distribution is less than or equal to a given value. In the video, the Z-table is used to find the probability that the sample mean of water consumption per person is more than 2.2 liters, which corresponds to running out of water.
Highlights

The average male requires 2 liters of water when active outdoors, with a standard deviation of 0.7 liters.

Planning a nature trip for 50 men with 110 liters of water brings up the question of water sufficiency.

The probability of running out of water is equated to the likelihood of average water consumption exceeding 2.2 liters per man.

A sampling distribution of the sample mean is introduced to determine the likelihood of water shortage.

The mean of the sampling distribution of the sample mean equals the population mean, which is 2 liters.

The standard deviation of the sampling distribution of the sample mean is calculated as the population standard deviation divided by the square root of the sample size.

For a sample size of 50, the standard deviation of the sample mean is approximately 0.099 liters.

The problem is reframed as finding the probability that the sample mean water consumption is greater than 2.2 liters.

A Z-score is calculated to determine how many standard deviations above the mean the critical value of 2.2 liters lies.

The Z-score calculation involves dividing the difference between the critical value and the mean by the standard deviation of the sampling distribution.

The Z-score for the critical value of 2.2 liters is approximately 2.02 standard deviations above the mean.

A Z-table is used to find the probability that the sample mean exceeds the critical value.

The probability of the sample mean being more than 2.02 standard deviations above the mean is found to be 0.9783 from the Z-table.

The final probability of running out of water is calculated by subtracting the Z-table probability from 1, resulting in 0.0217 or 2.17%.

The conclusion is that there is a 2.17% chance of running out of water during the planned nature trip.

Transcripts

Browse More Related Video

Calculating Power and the Probability of a Type II Error (A One-Tailed Example)

Math 14 HW 6.4.6-T Using the Central Limit Theorem

The Probability an Elevator is Overloaded Central Limit Theorem with StatCrunch

Standard error of the mean | Inferential statistics | Probability and Statistics | Khan Academy

SAMPLING DISTRIBUTION OF SAMPLE MEANS - WITH AND WITHOUT REPLACEMENT

Math 14 HW 6.4.15-T Using the Central Limit Theorem

Rate This

5.0 / 5 (0 votes)

Thanks for rating:

Related Tags
Probability CalculationStatistical AnalysisWater ConsumptionOutdoor TripStandard DeviationSample MeanNormal DistributionZ-ScoreRisk AssessmentHiking SafetyStatistical Literacy