Normal Distribution: Calculating Quantiles {TI 84 Plus CE}

Bell Curved Education
6 May 201705:06
EducationalLearning
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TLDRThis video script explains how to use a graphing calculator to find the cutoff scores for schools in a contest, based on statistical distributions. It demonstrates finding the minimum score needed to be in the top 70% and top 10% using the inverse norm function, with a normal distribution graph having a mean of 11 and a standard deviation of 3. The process involves calculating the K value for the desired percentile, taking into account the calculator's method of finding areas to the left of K. The video provides step-by-step instructions for using the TI-84 Plus CE calculator to solve these problems.

Takeaways
  • ๐Ÿ“ˆ The video demonstrates a normal distribution graph with a mean of 11 and a standard deviation of 3.
  • ๐Ÿ† It presents a scenario where schools are in a contest, and to advance, they need to be in the top 70%, implying a cutoff at the bottom 30%.
  • ๐Ÿ” The goal is to find the value 'K' such that the area under the normal curve to the left of 'K' is 30% or 0.3.
  • ๐Ÿงฎ The video explains using a graphing calculator to find 'K' by setting the probability of an 'x' value being less than 'K' to 0.3.
  • ๐Ÿ“ฑ The TI-84 Plus calculator is used to find 'K' by using the inverse norm function, which calculates the area to the left of 'K'.
  • ๐Ÿ“Š The calculator is instructed to find the area to the left of 'K' with a probability of 0.3, using the given mean and standard deviation.
  • ๐Ÿค” The video then shifts focus to finding the top 10%, which requires a different approach to calculate 'K'.
  • ๐Ÿ”„ For the top 10%, the calculator must find the area to the left of 'K' that is not 10%, but instead 90%, by using 1 - 0.1.
  • ๐Ÿ“ The video emphasizes that the TI-84 Plus calculator only finds the area to the left of 'K', so the question must be rephrased to find the correct 'K' value.
  • ๐ŸŽฏ The final 'K' values are approximately 9.43 for the bottom 30% and 14.84 for the top 10% of the distribution.
  • ๐Ÿ“š The key takeaway is understanding how to use the inverse function of the calculator to find the correct 'K' value based on the desired percentile.
Q & A

  • What is the mean of the normal distribution graph presented in the video?

    -The mean of the normal distribution graph is eleven.

  • What is the standard deviation of the normal distribution graph in the video?

    -The standard deviation is three.

  • What does the video suggest is necessary for a school to advance to the next round in the contest?

    -To advance to the next round, a school must be in the top 70%, which means it should not be in the bottom 30%.

  • What is the value 'K' in the context of the video?

    -The value 'K' represents the score or point on the normal distribution graph where the cumulative area to the left of 'K' is 30% or 0.3, indicating the bottom 30% of schools.

  • How can one find the value 'K' using a graphics calculator?

    -To find 'K', you can use the inverse norm function on a graphics calculator, inputting the desired area (0.3 for the bottom 30%), the mean (11), and the standard deviation (3).

  • What is the approximate value of 'K' for the bottom 30% as calculated in the video?

    -The approximate value of 'K' for the bottom 30% is 9.43.

  • What if the interest is in finding the top 10% instead of the bottom 30%?

    -If you are interested in the top 10%, you need to find the K value for which the area to the left is 90% (1 - 0.1), because the calculator finds the area to the left of K.

  • What is the approximate value of 'K' for the top 10% as calculated in the video?

    -The approximate value of 'K' for the top 10% is 14.84.

  • Why do you need to adjust the question when looking for the top percentage using the calculator?

    -You need to adjust the question because the calculator finds the area to the left of 'K', so for the top percentage, you calculate the complement to 1 and find the area to the left for that complement.

  • What is a key takeaway from the video regarding the use of the inverse function on a TI-84 Plus CE calculator?

    -The key takeaway is that the inverse function on the TI-84 Plus CE calculator only finds the area to the left of 'K', so you must reframe the question to find the correct K value for the desired percentile.

Outlines
00:00
๐Ÿ“Š Understanding Normal Distribution and Percentiles

This paragraph introduces a normal distribution graph with a mean of 11 and a standard deviation of 3, using a hypothetical scenario of schools in a contest. The goal is to determine the minimum score (K value) that places a school in the top 70%, thus advancing to the next round. The explanation involves finding the probability that an x-value is less than K, where the area under the curve to the left of K is 30% or 0.3. The TI-84 Plus calculator is used to find this K value by using the inverse norm function, which calculates the area to the left of K for a given probability. The process is demonstrated with the calculator, yielding an approximate K value of 9.43 for the top 70%.

Mindmap
Keywords
๐Ÿ’กNormal Distribution
Normal distribution, also known as Gaussian distribution, is a probability distribution that is characterized by its symmetric bell-shaped curve. In the video, the normal distribution graph represents the performance of schools in a contest, with the mean and standard deviation being key parameters that define the distribution's shape and spread. The mean is set at eleven, indicating the average performance, while the standard deviation of three shows the variability of the scores around the mean.
๐Ÿ’กMean
The mean, often referred to as the average, is a measure of central tendency in a set of numbers. In the context of the video, the mean of the normal distribution graph is eleven, which signifies the average score of the schools. It is a crucial value for understanding where the central mass of the data lies and serves as a reference point for determining other statistics such as percentiles.
๐Ÿ’กStandard Deviation
Standard deviation is a measure that indicates the amount of variation or dispersion in a set of values. With a standard deviation of three in the video's normal distribution, it tells us how spread out the school scores are from the mean. A higher standard deviation would indicate a greater spread, while a lower one would suggest scores are closer to the mean.
๐Ÿ’กTop 70%
The term 'top 70%' refers to the highest performing segment of the schools in the contest, which in this case is 70% of all participants. The video discusses how to determine the minimum score (K value) needed for a school to be in this top 70% to advance to the next round, emphasizing the importance of identifying the cutoff point for this percentile.
๐Ÿ’กCutoff Point
A cutoff point is a threshold value that separates one group from another based on a specific criterion. In the script, the cutoff point is used to determine the minimum score required for schools to be in the top 70% and, separately, the top 10%. It is the value of K that delineates the performance levels for advancement in the contest.
๐Ÿ’กInverse Norm
The inverse norm, or the quantile function, is used to find the value of a random variable for a given probability. In the video, the inverse norm function on a graphics calculator is utilized to calculate the K value corresponding to the 30% and 10% areas under the normal distribution curve. It's a method to solve for the score that corresponds to a specific percentile.
๐Ÿ’กProbability
Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. The video script discusses finding the probability that an x-value (school score) is less than K, which is equivalent to finding the area under the normal distribution curve to the left of K. This probability is set to 0.3 for the top 70% and adjusted for the top 10% scenario.
๐Ÿ’กGraphical Calculator
A graphical calculator, like the one mentioned in the video, is a device that can perform complex mathematical calculations and graph functions. In this context, it is used to find the K value for specific probabilities within the normal distribution. The calculator's 'inverse norm' function is a key tool for determining the cutoff scores for the contest.
๐Ÿ’กArea Under the Curve
The area under the curve in a probability distribution represents the probability of a random variable falling within a certain range. In the video, the area under the curve to the left of K is calculated to be 0.3 for the top 70% and 0.9 for the remaining 90% when looking for the top 10%. This area is crucial for determining the percentile cutoffs.
๐Ÿ’กPercentile
A percentile is a value below which a given percentage of observations in a group of observations fall. The video discusses determining the scores that correspond to the 70th and 10th percentiles of the school contest scores. These percentiles help in identifying the minimum performance levels required to be in the top segments of the contest.
Highlights

Introduction of a normal distribution graph with a mean of 11 and a standard deviation of 3.

Explanation of a contest scenario where schools need to be in the top 70% to advance.

The goal is to find the value K, where the area under the curve to the left of K is 30%.

Using a graphics calculator to find the value K for a given probability.

Navigating to the 'inverse norm' function on the calculator to solve for K.

Entering the area to the left of K as 0.3 (30%) to find the minimum score for advancement.

Inputting the mean (11) and standard deviation (3) into the calculator.

Revealing the calculated K value of approximately 9.43 for the top 70%.

Exploring the scenario of finding the top 10% of scores.

Understanding that the calculator finds the area to the left of K, requiring a reverse approach for top percentages.

Adjusting the problem to find the area to the left of K as 90% to solve for the top 10%.

Entering the adjusted probability of 0.9 into the calculator for the top 10% scenario.

Calculating the new K value for the top 10%, which is approximately 14.84.

Emphasizing the importance of understanding how the calculator finds areas to the left of K.

Highlighting the need to adjust the question when looking for top percentages to find the correct K value.

Concluding with a summary of the process and the importance of correctly using the calculator's inverse function.

Transcripts

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Normal DistributionTI-84CalculatorPercentilesStatistical ContestTop 70%Bottom 30%Top 10%Inverse NormEducational MathProbability