6.12
Anwendungen der Normalverteilung
Die Normalverteilung ist ein nützliches statistisches Werkzeug. Eine ihrer praktischen Anwendungen ist die Bestimmung der Türhöhe, nachdem die Normalv…
Die Normalverteilung ist auf viele Probleme im wirklichen Leben anwendbar.
Zum Beispiel werden die Statistiken der menschlichen Körpergröße verwendet, um die Türhöhe zu bestimmen, die es der Mehrheit der Menschen ermöglicht, hindurchzugehen, ohne sich den Kopf zu stoßen.
Nehmen wir an, dass der Mensch eine mittlere Körpergröße von 1,7 Metern mit einer Standardabweichung von 0,06 Metern hat.
Der schattierte Bereich in der Normalverteilung stellt Menschen dar, die 1,9 Meter oder weniger groß sind.
Konvertieren Sie zunächst die Zufallsvariable auf der X-Achse in Z-Werte, um eine Standardnormalverteilung zu erhalten.
Eine Höhe von 1,9 Metern entspricht einem Z-Score von 3,33. Die entsprechende Wahrscheinlichkeit wird in der z-Score-Tabelle nachgeschlagen.
Die Wahrscheinlichkeit liegt bei 0,9996, was uns sagt, dass 99,96 Prozent der Menschen durch eine 1,9 Meter hohe Tür gehen können.
In ähnlicher Weise können wir die Türhöhe berechnen, die es mindestens 85 % der Personen ermöglichen würde, ohne sich zu bücken.
Notieren Sie sich in der z-Tabelle den Wert des z-Werts für eine Wahrscheinlichkeit von 0,85.
Mit diesem z-Score wird die benötigte Türhöhe berechnet.
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Q1: How is the normal distribution used to determine appropriate door heights?
The normal distribution models human height variation to design doors that accommodate most people safely. Given a mean height of 1.7 meters and standard deviation of 0.06 meters, a 1.9-meter door allows 99.96% of people to pass through without striking their heads. This practical application uses z scores and area under the curve to calculate the probability that individuals can safely use the doorway.
Q2: What is a z score and how does it help analyze height data?
A z score standardizes individual measurements by converting them to a common scale relative to the mean and standard deviation. For example, a height of 1.9 meters with mean 1.7 and standard deviation 0.06 yields a z score of 3.33. This conversion allows researchers to use standard normal distribution tables to find probabilities and make comparisons across different datasets.
Q3: How can you find the door height that allows 85% of people to pass through?
Identify the z score corresponding to a cumulative probability of 0.85 from the z score table. Then use the z score formula in reverse: height equals mean plus z score multiplied by standard deviation. This calculation determines the specific door height needed to accommodate at least 85% of the population without requiring them to bend.
Q4: What does a z score table tell you about population probabilities?
A z score table provides cumulative probabilities showing the proportion of a population falling below any given z score value. For instance, a z score of -1.5 corresponds to a probability of 0.0668, meaning 6.68% of the population falls below that point. This allows statisticians to quantify how common or rare specific measurements are within a normally distributed population.
Q5: How was the normal distribution applied to Chilean male height data?
Heights of 15 to 18-year-old males from Chile (1984-1985) followed a normal distribution with mean 172.36 cm and standard deviation 6.34 cm. Using z scores, researchers calculated that 6.68% of males had heights below 162.85 cm. This demonstrates how normal distribution analysis quantifies the probability of specific height ranges in real populations.
Q6: Why is converting raw measurements to z scores important for probability calculations?
Converting measurements to z scores standardizes data to a common scale, enabling use of universal z score tables for probability lookup. Without this conversion, each dataset would require its own probability table. Z scores allow researchers to compare measurements across different populations and quickly determine cumulative probabilities for any normally distributed variable.
Q7: What real-world problems can be solved using normal distribution analysis?
Normal distribution solves practical problems involving population measurements, such as designing infrastructure (door heights), assessing health risks, or setting manufacturing tolerances. By analyzing how measurements like height, weight, or test scores distribute across populations, organizations can make evidence-based decisions that accommodate the majority while identifying unusual cases.