Consumption of alcoholic beverages by young women of...

Consumption of alcoholic beverages by young women of drinking age has been increasing in the United Kingdom, the United States, and Europe ( The Wall Street Journal, February 15,2006 ). Data (annual consumption in liters) consistent with the findings reported in The Wall Street Journal article are shown for a sample of 20 European young women. $$ \begin{array}{rrrrr} 266 & 82 & 199 & 174 & 97 \\ 170 & 222 & 115 & 130 & 169 \\ 164 & 102 & 113 & 171 & 0 \\ 93 & 0 & 93 & 110 & 130 \end{array} $$ Assuming the population is roughly symmetric, construct a $95 \%$ confidence interval for the mean annual consumption of alcoholic beverages by European young women.

Consumption of alcoholic beverages by young women of drinking age has been increasing in the United Kingdom, the United States, and Europe ( The Wall Street Journal, February 15,2006 ). Data (annual consumption in liters) consistent with the findings reported in The Wall Street Journal article are shown for a sample of 20 European young women.
$$
\begin{array}{rrrrr}
266 & 82 & 199 & 174 & 97 \\
170 & 222 & 115 & 130 & 169 \\
164 & 102 & 113 & 171 & 0 \\
93 & 0 & 93 & 110 & 130
\end{array}
$$
Assuming the population is roughly symmetric, construct a $95 \%$ confidence interval for the mean annual consumption of alcoholic beverages by European young women.

Key Concepts

Assumption of Normality

The assumption of normality, or at least symmetry in the population distribution, is important for justifying the use of the t-distribution in constructing confidence intervals, especially with small sample sizes. This assumption ensures that the sampling distribution of the mean is approximately normal, allowing for accurate inference using the t-distribution.

Standard Error

The standard error is a measure of the dispersion or variability of a sample statistic, most commonly the sample mean. It is calculated by dividing the sample standard deviation by the square root of the sample size, and it serves to quantify the precision of the sample estimate relative to the true population parameter.

Confidence Interval

A confidence interval is a statistical tool that provides a range of values within which the true population parameter, such as the mean, is expected to lie with a specified level of confidence (e.g., 95%). It accounts for variability in sample estimates and quantifies the uncertainty inherent in using a sample to estimate a population statistic.

t-Distribution

The t-distribution is used in inferential statistics when the sample size is small and/or the population variance is unknown. It is similar to the normal distribution but has heavier tails, which compensates for the additional uncertainty when estimating the standard deviation from a small sample.

Degrees of Freedom

Degrees of freedom refer to the number of independent pieces of information available to estimate another parameter. In the context of constructing a confidence interval using the t-distribution, the degrees of freedom are typically equal to the sample size minus one, which influences the exact form of the t-distribution used.

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