Math 14 6.2.25 Find the percentage of women meeting the height requirement.

Fiorentino Siciliano
13 Mar 202313:36
EducationalLearning
32 Likes 10 Comments

TLDRThis video script discusses a statistical analysis of women's heights in relation to military recruitment standards. It explains how to calculate the percentage of women meeting a height requirement using the normal distribution with a mean of 63.8 inches and a standard deviation of 2.4 inches. The script demonstrates finding Z-scores for the given height range and using them to determine that 99.22% of women qualify. It further explores adjusting height requirements to include all but the shortest 1% and tallest 2%, calculating new minimum and maximum height limits of 58.2 inches and 68.7 inches, respectively.

Takeaways
  • 📊 A survey reveals that women's heights are normally distributed with a mean of 63.8 inches and a standard deviation of 2.4 inches.
  • 🎯 The military branch has a height requirement of 58 to 80 inches for women to join.
  • 🔍 Part A of the question asks to find the percentage of women meeting the height requirement.
  • 📈 A bell curve is drawn to visualize the distribution and identify the range of interest (58 to 80 inches).
  • ⚖️ Z-scores are calculated for both 58 inches (X1) and 80 inches (X2) to standardize the distribution.
  • 🔢 The z-score for 58 inches is approximately -2.42, and for 80 inches, it is approximately 6.75.
  • 🤔 The percentage of women meeting the height requirement is calculated using the Z-scores and a standard normal distribution table or calculator.
  • 📉 The result shows that 99.22% of women meet the height requirement, indicating only a small percentage are not eligible.
  • 🚫 The script suggests that not many women are denied the opportunity to join due to height restrictions.
  • 🆕 Part B considers a change in height requirements to include all but the shortest 1% and tallest 2% of women.
  • 📐 New Z-scores are determined for the revised height requirements, which are -2.33 for the shortest 1% and 2.05 for the tallest 2%.
  • 📝 The new height requirements are calculated to be at least 58.2 inches and at most 68.7 inches, rounded to one decimal place.
Q & A

  • What is the mean height of women as per the survey?

    -The mean height of women according to the survey is 63.8 inches.

  • What is the standard deviation of women's heights in the survey?

    -The standard deviation of women's heights is 2.4 inches.

  • What is the height requirement range for a branch of the military mentioned in the script?

    -The military branch requires women's heights to be between 58 inches and 80 inches.

  • What percentage of women meet the height requirement for the military branch?

    -99.22 percent of women meet the height requirement for the military branch.

  • Are many women being denied the opportunity to join the military branch due to their height?

    -No, only a small percentage of women are not allowed to join due to their height, as 99.22 percent meet the requirement.

  • What is the purpose of finding the Z-scores for 58 inches and 80 inches?

    -The Z-scores for 58 inches and 80 inches are used to calculate the probability or percentage of women within that height range, which is necessary to determine the percentage meeting the military's height requirement.

  • What does the Z-score of -2.42 represent in the context of the script?

    -The Z-score of -2.42 represents the number of standard deviations 58 inches is below the mean height of 63.8 inches.

  • What does the Z-score of 6.75 signify?

    -The Z-score of 6.75 signifies that 80 inches is 6.75 standard deviations above the mean height of 63.8 inches.

  • How does the script use technology to find the probability between two Z-scores?

    -The script uses StatCrunch, a statistical software, to find the probability or area under the normal distribution curve between the two Z-scores of -2.42 and 6.75.

  • What is the new height requirement if the military branch changes it to exclude only the shortest 1% and tallest 2% of women?

    -The new height requirements would be at least 58.2 inches and at most 68.7 inches, excluding only the shortest 1% and tallest 2% of women.

  • How are the new height requirements calculated in the script?

    -The new height requirements are calculated by finding the Z-scores corresponding to the areas of 0.0100 (shortest 1%) and 0.0200 (tallest 2%), and then converting these Z-scores back to height values using the mean and standard deviation.

Outlines
00:00
📊 Calculating Women's Height Eligibility in the Military

This paragraph discusses a statistical approach to determine the percentage of women meeting a military branch's height requirements. The women's heights are normally distributed with a mean of 63.8 inches and a standard deviation of 2.4 inches. The military requires heights between 58 and 80 inches. The process involves calculating Z-scores for the lower and upper limits of the requirement and then finding the probability or area under the normal distribution curve between these Z-scores. The Z-scores for 58 and 80 inches are -2.42 and 6.75, respectively. Using a statistical tool like StatCrunch, the probability between these Z-scores is found to be 0.9922, or 99.22% when converted to a percentage, indicating that only a small percentage of women are denied the opportunity to join due to height restrictions.

05:01
🏆 Adjusting Military Height Requirements

The second paragraph continues the discussion on military height requirements but introduces a hypothetical scenario where the military changes its policy to include all women except for the shortest 1% and the tallest 2%. To find the new height requirements, the paragraph explains the process of determining the Z-scores that correspond to the areas representing the shortest 1% and tallest 2% of the population. Using StatCrunch, the Z-scores for these areas are found to be -2.33 and 2.05, respectively. These Z-scores are then used to calculate the new height limits by applying the formula for converting Z-scores back to the original scale of the data, resulting in a new minimum height requirement of 58.2 inches and a maximum of 68.7 inches.

10:03
🔢 Finalizing New Height Requirements for Military Enlistment

The final paragraph wraps up the discussion by summarizing the new height requirements for women wishing to join the military branch. It reiterates the process of converting Z-scores to actual height values and provides the rounded figures for the new minimum and maximum heights, which are 58.2 inches and 68.7 inches, respectively. This adjustment ensures that 99% of women are eligible for enlistment, significantly reducing the number of women denied the opportunity due to height restrictions.

Mindmap
Keywords
💡Normal Distribution
Normal distribution, also known as Gaussian distribution, is a probability distribution that is characterized by its symmetrical bell-shaped curve. In the context of the video, the heights of women are normally distributed with a mean of 63.8 inches and a standard deviation of 2.4 inches, which means that the majority of women's heights will fall around the mean, with fewer individuals at the extremes.
💡Mean
The mean, often referred to as the average, is a measure of central tendency in statistics. It is calculated by adding all the values in a data set and then dividing by the number of values. In the video, the mean height of women is given as 63.8 inches, which is the central point around which the heights of women are distributed.
💡Standard Deviation
Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values are close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In the script, the standard deviation of women's heights is 2.4 inches, indicating the spread of height values around the mean.
💡Z-Score
A Z-score is a statistical measurement that indicates how many standard deviations an element is from the mean. In the video, Z-scores are calculated for the heights of 58 inches and 80 inches to determine how many standard deviations these heights are from the mean height of 63.8 inches. This helps in understanding the probability of women falling within certain height ranges.
💡Probability
Probability is a measure of the likelihood that a particular event will occur. The video discusses finding the probability of women's heights falling between 58 and 80 inches by calculating the area under the normal distribution curve between these two Z-scores, which represents the percentage of women meeting the height requirement.
💡Bell Curve
The bell curve is a graphical representation of the normal distribution, showing that data points are clustered around the mean with a symmetrical fall-off towards the extremes. In the video, the bell curve is used to visualize the distribution of women's heights and to identify the range of heights that meet the military's requirements.
💡Military Height Requirement
The military height requirement refers to the minimum and maximum height limits set by a branch of the military for eligibility. In the video, the initial requirement is between 58 and 80 inches, and the script explores the percentage of women who meet these criteria and the implications of changing these requirements.
💡Percentage
Percentage is a way of expressing a number as a fraction of 100. It is used in the video to express the proportion of women who meet the height requirements, with the initial calculation showing that 99.22% of women are eligible based on the given height range.
💡StatCrunch
StatCrunch is a statistical software tool used for data analysis, including the calculation of probabilities and Z-scores. In the script, StatCrunch is used to find the probabilities associated with the Z-scores for the height requirements, helping to determine the percentage of women who would be eligible to join the military.
💡Eligibility
Eligibility refers to the qualification or fitness to meet certain criteria or requirements. The video discusses changing the height requirements to make all women eligible except for the shortest 1% and the tallest 2%, using Z-scores to determine the new height limits for eligibility.
Highlights

A survey found women's heights are normally distributed with a mean of 63.8 inches and standard deviation of 2.4 inches.

A military branch requires women's heights to be between 58 and 80 inches.

Part A asks to find the percentage of women meeting the height requirement.

Only a small percentage of women are denied joining the military due to height.

A bell curve is drawn to visualize the distribution and calculate probabilities.

Z-scores are calculated for the heights 58 and 80 inches to find the probability between them.

The z-score for 58 inches is -2.42 and for 80 inches is 6.75.

Using a standard normal distribution table or calculator, the probability between z-scores -2.42 and 6.75 is found to be 99.22%.

99.22% of women meet the height requirement, with only a small percentage not eligible.

Part B asks for new height requirements excluding the shortest 1% and tallest 2% of women.

The shortest 1% and tallest 2% correspond to areas of 0.0100 and 0.0200 under the standard normal curve.

Z-scores for the shortest 1% and tallest 2% are calculated as -2.33 and 2.05 respectively.

Using the mean and standard deviation, the new height requirements are calculated from the z-scores.

The new minimum height requirement is 58.2 inches, calculated from the z-score -2.33.

The new maximum height requirement is 68.7 inches, calculated from the z-score 2.05.

The new requirements allow 98% of women to be eligible, excluding only the shortest 1% and tallest 2%.

Transcripts

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Related Tags
Statistical AnalysisWomen's HeightsMilitary EligibilityRecruitment StandardsNormal DistributionZ-Score CalculationHeight RequirementsData InterpretationProbability CalculationMilitary RecruitmentHealth Statistics