Question

The article "The Distribution of Buying Frequency Rates" (Journal of Marketing Research \([1980]: 210-216)\) reported the results of a \(3 \frac{1}{2}\) -year study of dentifrice purchases. The investigators conducted their research using a national sample of 2071 households and recorded the number of toothpaste purchases for each household participating in the study. The results are given in the following frequency distribution: $$ \begin{array}{cc} \begin{array}{l} \text { Number of } \\ \text { Purchases } \end{array} & \begin{array}{l} \text { Number of House- } \\ \text { holds (Frequency) } \end{array} \\ \hline 10 \text { to }<20 & 904 \\ 20 \text { to }<30 & 500 \\ 30 \text { to }<40 & 258 \\ 40 \text { to }<50 & 167 \\ 50 \text { to }<60 & 94 \\ 60 \text { to }<70 & 56 \\ 70 \text { to }<80 & 26 \\ 80 \text { to }<90 & 20 \\ 90 \text { to }<100 & 13 \\ 100 \text { to }<110 & 9 \\ 110 \text { to }<120 & 7 \\ 120 \text { to }<130 & 6 \\ 130 \text { to }<140 & 6 \\ 140 \text { to }<150 & 3 \\ 150 \text { to }<160 & 0 \\ 160 \text { to }<170 & 2 \\ \hline \end{array} $$ a. Draw a histogram for this frequency distribution. Would you describe the histogram as positively or negatively skewed? b. Does the square-root transformation result in a histogram that is more symmetric than that of the original data? (Be careful! This one is a bit tricky, because you don't have the raw data; transforming the endpoints of the class intervals will result in class intervals that are not necessarily of equal widths, so the histogram of the transformed values will have to be drawn with this in mind.)

Short answer

The solution involves drawing two histograms, one for the original data and the other for the square-root transformed data, as well as describing the skewness and symmetry for both. To provide an exact answer, these steps need to be followed out graphically.

Step-by-step solution

Draw Histogram for the Original Data

Start by using the Number of Purchases as intervals on the x-axis, and the Number of Households (Frequency) on the y-axis. Next, form the bars for each interval where the width is the purchase interval and the height is the corresponding frequency. Observe the shape to determine skewness.

Interpret Skewness

If the right (positive) side of the histogram extends more than the left (negative) side, the histogram is positively skewed. If the left (negative) side extends more than the right (positive) side, it is negatively skewed.

Apply and Draw Histogram for the Square-root Transformed Data

Apply the square-root transformation to each class interval (the 'number of purchases'). The resulting transformed values should be used as intervals on the x-axis. Note that class intervals are not necessarily of equal widths anymore when creating the histogram with the heights still corresponding to the frequencies.

Compare Symmetry

Observe the shape of the histogram to determine whether the transformed data histogram is more symmetric than that of the original data - is the distribution is more evenly spread around the median?