Empirical rule applies

Sample Solution

Height Distribution in Adults: A Normally Distributed Phenomenon

Data Collection and Empirical Rule:

One situation where you can collect data and expect it to follow the empirical rule (or the 68-95-99.7 rule) is the distribution of adult heights within a specific population. This means that the majority of adults will fall within a predictable range around the average height, with progressively fewer people falling outside this range.

Why Normal Distribution?

Several factors lead us to believe that adult height data follows a normal distribution:

• Multiple Contributing Factors: Height is influenced by a combination of genetics, nutrition, and environmental factors. These factors typically have a random, additive effect, which often results in a normal distribution according to the Central Limit Theorem [Source: National Institute of Standards and Technology (NIST), "Central Limit Theorem"].
• Large Sample Size: When measuring the heights of a large population sample, minor variations in individual factors tend to average out, leading to a bell-shaped curve.
• Historical Data: Extensive historical data on adult height distribution across various populations shows a strong tendency towards normality [Source: Our World in Data, "Global Height Trends"].

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Benefits of Normal Distribution:

Having normally distributed data on adult height offers several statistical analysis benefits:

• Predictability: The empirical rule allows us to estimate the percentage of people who fall within a certain height range based on the average and standard deviation.
• Hypothesis Testing: Many statistical tests, such as t-tests and z-tests, rely on the assumption of normality. Knowing the data is normally distributed simplifies hypothesis testing about the population mean or comparing means between groups.
• Efficiency in Sampling: When sampling a population to estimate average height, normal distribution allows us to determine a smaller sample size that can still provide reliable results.

Conclusion:

By collecting data on adult heights within a large population, we can expect it to follow a normal distribution. This knowledge allows for efficient data analysis, facilitates hypothesis testing, and enables predictions about the distribution of heights within the population.