normalCDF & InvNorm on TI-84 Plus Calculator: Normal Calculations

MATHRoberg
25 Feb 201705:16
EducationalLearning
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TLDRThis video script offers a detailed tutorial on using the normal distribution to solve problems related to gestation periods, specifically for pygmy hippos. It demonstrates how to find the area under the curve corresponding to a given cutoff value using the normal CDF function and how to determine the cutoff value for a specified area using the inverse norm function. The script also explains how to calculate the middle 70% range for gestation periods, providing step-by-step instructions on using a calculator for these statistical calculations.

Takeaways
  • ๐Ÿ“Š The gestation period of pygmy hippos is normally distributed with a mean of 200 days and a standard deviation of 3.5 days.
  • ๐Ÿ” To find the proportion of pregnancies lasting longer than a certain number of days, use the normal CDF (Cumulative Distribution Function) on a calculator.
  • ๐Ÿ“ The script demonstrates estimating the area under the normal curve by 'eyeballing', suggesting an area of about 10% for pregnancies lasting longer than 204 days.
  • ๐Ÿงฎ For calculating the exact area, input the lower limit (cutoff value) and set the upper limit to positive infinity in the normal CDF function.
  • ๐Ÿ”‘ The normal CDF function requires the mean (200 days) and standard deviation (3.5 days) to calculate the area.
  • ๐Ÿ” Inverse norm is used when you have an area and want to find the corresponding cutoff value.
  • ๐Ÿ“ˆ The inverse norm function on a calculator helps determine the cutoff value that corresponds to a specific area to the left of it.
  • ๐Ÿ“Š The script shows how to find the cutoff value for the lower 10% of pregnancies by using the inverse norm with an area of 10%.
  • ๐Ÿ”ข The cutoff value for the lower 10% of pregnancies is approximately 195.5 days.
  • ๐Ÿ“‰ To find the cutoff for the upper 20% of pregnancies, use the inverse norm with an area of 0.8 (since 20% is to the right of the cutoff).
  • ๐Ÿ“‹ The script concludes with finding the middle 70% of pregnancies, which involves calculating two cutoff values for the lower and upper tails, resulting in a range of 196.4 to 200.36 days.
  • ๐Ÿ“š The video recommends a book titled 'The Ultimate AP Statistics Practice Book' for further practice with similar problems.
Q & A

  • What distribution is assumed for the gestation period in the script?

    -The gestation period is assumed to be normally distributed in the script.

  • What is the mean of the normal distribution for the gestation period?

    -The mean of the normal distribution for the gestation period is 200 days.

  • What is the standard deviation of the gestation period in the script?

    -The standard deviation of the gestation period is 3.5 days.

  • How is the proportion of gestation periods longer than 204 days calculated in the script?

    -The proportion is estimated by using the normal CDF function on a calculator, with the lower limit set to 204 and the upper limit set to positive infinity.

  • What function on the calculator is used to find the area when you have a cutoff value?

    -The normal CDF (Cumulative Distribution Function) is used to find the area when you have a cutoff value.

  • How do you determine the cutoff value when you know the area?

    -You use the inverse norm function on the calculator to determine the cutoff value when you know the area.

  • What is the cutoff value for the lower 10% of hippo pregnancies in the script?

    -The cutoff value for the lower 10% of hippo pregnancies is approximately 195.5 days.

  • What is the cutoff value for the upper 20% of pygmy hippo pregnancies in the script?

    -The cutoff value for the upper 20% of pygmy hippo pregnancies is approximately 200.29 days.

  • How do you calculate the middle 70% of the gestation period in the script?

    -You calculate the middle 70% by finding two cutoff values using the inverse norm function for the areas 0.15 and 0.85, which correspond to the tails of the distribution.

  • What are the lower and upper cutoff values for the middle 70% of pygmy hippo pregnancies in the script?

    -The lower cutoff value for the middle 70% of pygmy hippo pregnancies is approximately 196.4 days, and the upper cutoff value is approximately 200.36 days.

  • What is the book recommended in the script for those interested in AP Statistics practice?

    -The book recommended in the script is 'The Ultimate AP Statistics Practice Book', which contains 100 problems with videos.

Outlines
00:00
๐Ÿ“Š Normal Distribution and Gestation Period Analysis

This paragraph explains the concept of a normal distribution in the context of gestation periods, using a mean of 200 and a standard deviation of 3.5. It demonstrates how to estimate the proportion of pregnancies lasting longer than 204 days by visually inspecting a normal distribution curve. The explanation includes using a calculator's normal CDF function to find the exact area under the curve, which is approximately 12.65%. The paragraph also covers how to use the inverse norm function to find the cutoff value when given an area, as illustrated with examples of finding the lower 10% and upper 20% cutoffs for pygmy hippo pregnancies.

05:05
๐Ÿ“š Ultimate AP Statistics Practice Book Promotion

The second paragraph serves as a promotional segment for an AP Statistics practice book that contains 100 problems with accompanying video explanations. The book is available for purchase on Amazon.com, suggesting that viewers who found the video content helpful might also benefit from the book for further practice and understanding of statistics problems.

Mindmap
Keywords
๐Ÿ’กGestation period
The gestation period refers to the time duration from conception to birth. In the context of the video, it is used to describe the variable time frame for pygmy hippo pregnancies, which is normally distributed with a mean of 200 days. The script discusses how to calculate the proportion of pregnancies lasting longer than a certain number of days using the normal distribution.
๐Ÿ’กNormal distribution
A normal distribution, also known as Gaussian distribution, is a probability distribution that is symmetric about the mean and follows a specific mathematical formula. The video uses the normal distribution to model the gestation periods of pygmy hippos, with the mean and standard deviation being key parameters that define the shape of the distribution.
๐Ÿ’กMean
The mean, often referred to as the average, is a measure of central tendency in a set of numbers. In the video, the mean gestation period of 200 days is the central point around which the normal distribution of pygmy hippo pregnancies is centered.
๐Ÿ’กStandard deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. With a standard deviation of 3.5 days for the gestation periods, the video illustrates how it affects the spread of the normal distribution, indicating that most pygmy hippo pregnancies will be close to the mean but within this range of variability.
๐Ÿ’กProportion
Proportion refers to a part of a whole, often expressed as a percentage or fraction. The video is focused on finding proportions of the gestation period that fall above or below certain thresholds, such as the percentage of pregnancies lasting longer than 204 days.
๐Ÿ’กArea under the curve
In the context of a normal distribution graph, the area under the curve represents the total probability and is always equal to one. The video discusses how to estimate or calculate the area under the curve to find the proportion of data points within a specific range, such as the proportion of pregnancies longer than 204 days.
๐Ÿ’กNormal CDF function
The Normal Cumulative Distribution Function (CDF) is used to calculate the probability that a normally distributed random variable is less than or equal to a certain value. In the script, the Normal CDF function is used with a calculator to find the area to the left of a given cutoff value, such as the proportion of pregnancies longer than 204 days.
๐Ÿ’กCutoff value
A cutoff value is a threshold used to separate data into different categories or to determine the likelihood of an event. The video script describes how to use the normal distribution to find the cutoff values that correspond to specific areas under the curve, such as the value that cuts off the top 20% of gestation periods.
๐Ÿ’กInverse norm
The inverse norm function, or inverse normal distribution function, is used to find the value from a given cumulative probability. In the video, the inverse norm is used to determine the cutoff gestation period that corresponds to a specific area, such as finding the value that cuts off the lower 10% of hippo pregnancies.
๐Ÿ’กUpper and lower tails
In a normal distribution graph, the upper and lower tails represent the areas to the right and left of the mean, respectively. The video discusses finding the cutoff values for the upper 20% and lower 10% of the distribution, which involves calculating the areas in these tails.
๐Ÿ’กMiddle 70%
The middle 70% of a distribution refers to the range of values that includes the central 70% of the data. The video script calculates the gestation period range that includes the middle 70% of pygmy hippo pregnancies, which involves finding two cutoff values that encompass this central area.
Highlights

Introduction of a normal distribution to model gestation periods with a mean of 200 and standard deviation of 3.5.

Visual estimation of the proportion of gestation periods exceeding 204 days, estimated to be around 10%.

Use of the normal CDF function on a calculator to find the area for a given cutoff value.

Explanation of how to input parameters into the calculator for the normal CDF function.

Calculation of the proportion of gestation periods longer than 204 days, approximately 12.65%.

Method to find the cutoff value when given an area using the inverse norm function on a calculator.

Guidance on using the inverse norm function with the calculator for determining the cutoff value.

Determining the cutoff value for the lower 10% of hippo pregnancies, approximately 195.5 days.

Illustration of finding the cutoff value for the upper 20% of gestation periods using inverse norm.

Calculation of the cutoff value for the upper 20% of gestation periods, approximately 200.29 days.

Explanation of finding the middle 70% of gestation periods by identifying two cutoff values.

Calculation of the lower cutoff value for the middle 70% of gestation periods, approximately 196.4 days.

Calculation of the upper cutoff value for the middle 70% of gestation periods, approximately 200.36 days.

Summary of the gestation period range for the middle 70% of pygmy hippo pregnancies.

Promotion of an AP Statistics practice book with video explanations similar to the content of the video.

Availability of the book on Amazon for those interested in further study and practice.

Transcripts

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StatisticsNormal DistributionGestation PeriodCalculusEducationalAP StatisticsVideo TutorialCDF FunctionInverse NormPractice Problems