6.12
正規分布の応用
正規分布は便利な統計ツールです。実際の応用例の 1 つは、人の身長の正規分布を考慮して、多くの人が頭をぶつけずに簡単に通過できるドアの高さを決定することです。正規分布は、特定の身長未満の身長の人の確率を求めることもできます。
1984年から1985年にかけてのチリの15歳から18歳の男性の身長は正規…
正規分布は、実際の多くの問題に広く適用できます。
たとえば、人間の身長の統計は、大多数の人々が頭を打たずに通り抜けることができるドアの高さを決定するために使用されます。
人間の平均身長が 1.7 メートル、標準偏差が 0.06 メートルであると仮定します。
正規分布の陰影領域は、身長が 1.9 メートル以下の人間を表します。
まず、X軸の確率変数をzスコアに変換して、標準正規分布を求めます。
1.9 メートルの高さは z スコア 3.33 に相当します。対応する確率は、zスコアテーブルで検索されます。
確率は0.9996で、99.96%の人が高さ1.9メートルのドアを通り抜けることができることがわかります。
同様に、少なくとも85%の人が曲がらずに通過できるドアの高さを計算できます。
z テーブルから、0.85 の確率の z スコアの値を確認します。
この z スコアを使用して、必要なドアの高さが計算されます。
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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.