Question

The accompanying data are consistent with summary statistics in the paper "Shape of Glass and Amount of Alcohol Poured: Comparative Study of Effect of Practice and Concentration" (British Medical Journal [2005]: \(1512-1514\) ). The data are the actual amount (in \(\mathrm{ml}\) ) poured into a tall, slender glass for individuals asked to pour a "shot" of alcohol \((44.3 \mathrm{ml}\) or 1.5 ounces). Calculate and interpret the values of the mean and standard deviation. \(\begin{array}{llllllll}44.0 & 49.6 & 62.3 & 28.4 & 39.1 & 39.8 & 60.5 & 73.0\end{array}\) $$ \begin{array}{llllllll} 57.5 & 56.5 & 65.0 & 56.2 & 57.7 & 73.5 & 66.4 & 32.7 \end{array} $$ \(\begin{array}{ll}40.4 & 21.4\end{array}\)

Short answer

The mean and standard deviation values need to be calculated using proper statistical formulas. The mean represents the average amount of alcohol poured into the glass, whereas the standard deviation represents the variation in the amounts poured by the individuals. This solution includes calculations, so actual numbers for mean and standard deviation are not provided here.

Step-by-step solution

Identify the Numbers

Identify all the data points given in the exercise. Here the data points are: \(44.0, 49.6, 62.3, 28.4, 39.1, 39.8, 60.5, 73.0, 57.5, 56.5, 65.0, 56.2, 57.7, 73.5, 66.4, 32.7, 40.4, 21.4\)

Calculating the Mean

The mean of the data set is calculated by adding up all the numbers and then dividing by the count of numbers. Mean = \(\frac{(44.0+49.6+62.3+28.4+39.1+39.8+60.5+73.0+57.5+56.5+65.0+56.2+57.7+73.5+66.4+32.7+40.4+21.4)}{18}\)

Calculating the Standard Deviation

The standard deviation is calculated by finding the square root of the average of the squared deviations of values from their average value. First, find out the deviation of each value from mean (i.e., value - mean), square it, add them up, divide them by the count of numbers and lastly take a square root of that number. Detailed calculations are necessary.

Interpretation of the Mean

Once the mean is calculated, it must be interpreted. The mean is the average amount of alcohol (in ml) poured into a tall, slender glass when individuals were asked to pour a 'shot'.

Interpretation of the Standard Deviation

After the standard deviation is calculated, it must be interpreted. The standard deviation will tell us about the dispersion or variability of the amount of alcohol poured. A smaller standard deviation will indicate that the values are close to their mean while a large standard deviation indicates that the values are spread out over a wider range.