Question

The data set SMOKE contains information on smoking behavior and other variables for a random sample of single adults from the United States. The variable cigs is the (average) number of cigarettes smoked per day. Do you think cigs has a normal distribution in the U.S. adult population? Explain.

Short answer

'Cigs' likely does not follow a normal distribution due to non-negative values and possible bimodality.

Step-by-step solution

Understand the Variable

The variable 'cigs' represents the average number of cigarettes smoked per day by single adults in the United States. The question asks about the distribution of this variable in the population.

Characteristics of a Normal Distribution

A normal distribution is symmetric about the mean, with most data points near the mean and fewer and fewer as one moves away. It has the bell-shaped curve, and its mean, median, and mode are all equal.

Examine the Nature of 'Cigs'

Consider that the average number of cigarettes smoked cannot be negative, which implies a lower bound at zero. It's also likely that there will be smokers and non-smokers, creating two peaks (bimodal distribution), one at zero for non-smokers and another peak for smokers at a positive average cigarette count, which is not typical of a normal distribution.

Assess Real-World Implications

In the real world, smoking habits can be influenced by many factors, meaning the distribution might be skewed or have outliers. Furthermore, the prevalence of non-smokers often results in many people at zero, adding more reasoning for a non-normal distribution.

Conclusion on Distribution

Based on the above considerations, it is unlikely that 'cigs' follows a normal distribution. Features such as a non-negative restriction and bimodality make 'cigs' inconsistent with the properties of a normal distribution.

Key concepts

Normal Distribution

A normal distribution is a common statistical concept that resembles a bell-shaped curve when graphed. This curve is perfectly symmetrical, meaning that if you were to cut it in half at the center, both sides would be mirror images of each other. The center of a normal distribution is characterized by three key statistical measures: the mean, median, and mode, all of which are equal.

In practical terms, the majority of data points in a normal distribution are clustered around the center mean, with fewer data points appearing as you move further from the center in either direction. This pattern assumes that data will have a similar frequency or occurrence on both sides of the average. When analyzing the distribution of cigarette smoking ("cigs") among adults in the U.S., you would expect a normal distribution to feature most adults smoking about the same number of cigarettes, with fewer individuals smoking an extremely low or high number.

• Symmetrical shape
• Mean = Median = Mode
• Most data near the mean

However, due to factors like the existence of non-smokers and varying smoking habits, this distribution might not be the best fit for understanding smoking behavior.

Bimodal Distribution

A bimodal distribution is one where two distinct groups appear within the data, resulting in two separate peaks within the distribution graph. This happens when there are two prevalent modes, or typical values, present in the data.

In the context of cigarette smoking habits, you might see one peak at zero cigarettes per day, representing non-smokers, and another peak at a positive number, indicative of those who do smoke. This distribution highlights a significant spli in the population: individuals who don't smoke and those who do, often with specific and varied smoking habits. Such bimodality suggests that the population cannot be adequately described by a single central point such as the mean.

• Two peaks in the data
• Reflects two groups or modes
• Ineffective single central point representation

This can serve as a critical insight for statisticians when modeling data, as it may indicate that more than one statistical strategy is necessary to fully understand the population.

Skewed Distribution

A skewed distribution is one where the symmetry found in a normal distribution is absent. Instead, one tail of the distribution stretches out longer than the other. This skew can be either to the right (positive skewness), where the tail goes further in the positive direction, or to the left (negative skewness), where it goes further in the negative direction.

When examining a skewed distribution in the smoking habits of adults, it's likely there will be a skew to the right. This means that while a significant number iof individuals may not smoke at all or smoke very few cigarettes, a smaller number may smoke a large amount, thus extending the right tail of the distribution. Understanding skewness is important because it affects data analysis, particularly measures of central tendency like the mean which can be heavily influenced by data outliers.

• Lack of symmetry
• Longer tail on one side
• Affects central tendency measures

For datasets like 'cigs', skewness provides invaluable insights into how spread out the data points are and the potential outliers that could influence overall trends.

Descriptive Statistics

Descriptive statistics are the summary statistics that quantitatively describe or summarize features from a collection of data. The goal of descriptive statistics is to provide a concise overview of the main characteristics of a dataset through summary measures like mean, median, mode, range, and standard deviation.

By applying these statistics to the 'cigs' dataset, one can quickly grasp important facts such as the average number of cigarettes smoked in a day, the most frequently occurring smoking rate, and how much variation exists in smoking habits across individuals. Descriptive statistics also help identify characteristics like skewness or bimodality, which indicates a need for more complex analyses or a consideration of multiple factors in the interpretation of data.

• Mean, median, mode
• Standard deviation
• Range

Using these tools can provide a powerful snapshot of the dataset's essential qualities, making it easier to understand complex and varied human behaviors like smoking.