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Chapter 4: Problem 40

The article "Taxable Wealth and Alcoholic Beverage Consumption in the United States" (Psychological Reports [1994]: \(813-814\) ) reported that the mean annual adult consumption of wine was \(3.15\) gal and that the standard deviation was \(6.09\) gal. Would you use the Empirical Rule to approximate the proportion of adults who consume more than \(9.24\) gal (i.e., the proportion of adults whose consumption value exceeds the mean by more than 1 standard deviation)? Explain your reasoning.

Short Answer

Expert verified
Under the assumption of a normal distribution, per the Empirical rule, approximately 68% of adults would consume \(3.15\) gal and \(9.24\) gal of wine annually. However, without the information confirming that the data is normally distributed, it's not definitive to apply the Empirical Rule.

Step by step solution

01

- Understand the Empirical Rule

The Empirical Rule in statistics states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean. Specifically, 68% of data falls within the first standard deviation (\(\mu 卤 \sigma\)), 95% within the first two standard deviations (\(\mu 卤 2\sigma\)), and 99.7% within three standard deviations (\(\mu 卤 3\sigma\)).
02

- Assess Distribution

Before we apply the Empirical Rule, it's important to know that it only applies to data that is normally distributed. This exercise does not provide any information about the distribution of data. So, we can't definitively say whether the data follows a normal distribution or not.
03

- Calculate quantity under consideration

The consumption value under consideration is \(9.24\) gal. Subtract the mean (\(3.15\) gal) from this value to get \(9.24 - 3.15 = 6.09\) gal which is equal to one standard deviation. This means we are looking for the proportion of data that falls within one standard deviation of the mean.
04

- Conclude

Assuming a normal distribution, we could use the Empirical Rule to say that approximately 68% of adults would consume between \(3.15\) gal and \(9.24\) gal of wine annually. However, we still can鈥檛 apply the Empirical Rule with certainty because we don鈥檛 have any information confirming that the data is normally distributed.

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