As part of a statistics project, Gill collected data relating to the length of time, to the nearest minute, spent by shoppers in a supermarket and the amount of money they spent - Edexcel - A-Level Maths Statistics - Question 1 - 2007 - Paper 1

To calculate $S_{tt}$, $S_{mm}$, and $S_{tm}$, we will use the provided sums:

• Calculate the mean values:

  • ar{t} = \frac{\sum{t}}{n} = \frac{212}{10} = 21.2
  • ar{m} = \frac{\sum{m}}{n} = \frac{61}{10} = 6.1

• Using the formulas:

  • $S_{tt} = \sum{t^2} - n \bar{t}^2 = 5478 - 10 \times (21.2)^2 = 5478 - 4504.8 = 973.2$
  • $S_{mm} = \sum{m^2} - n \bar{m}^2 = 2101 - 10 \times (6.1)^2 = 2101 - 372.1 = 1728.9$
  • $S_{tm} = \sum{tm} - n \bar{t} \bar{m} = 2485 - 10 \times 21.2 \times 6.1 = 2485 - 1293.2 = 1191.8$.

The product moment correlation coefficient can be calculated using the formula: $r = \frac{S_{tm}}{\sqrt{S_{tt} \cdot S_{mm}}}$ Substituting the values found: $r = \frac{1191.8}{\sqrt{973.2 \cdot 1728.9}} \approx 0.914$.

The product moment correlation coefficient between $t$ and the actual amount spent is $r = 0.178$. This indicates a weak positive correlation, suggesting that as the time spent increases, the actual amount spent does not change significantly. This may be due to customers spending time on activities that don’t lead to increased spending.

The coefficient of $0.914$ implies a strong positive correlation between time spent in the supermarket and the amount spent, indicating that shoppers who spend more time tend to spend more money. On the other hand, the coefficient of $0.178$ indicates a weak correlation suggesting that time spent does not necessarily translate to higher spending, possibly because other factors (like shopping habits) play a considerable role.

One practical reason for the difference between the two correlation coefficients could be related to external factors influencing shopper behavior, such as the day of the week or time of day. For example, shoppers may spend more time in the store on weekends, but their spending might not increase proportionately due to other distractions or purchasing decisions influenced by special promotions or the presence of sales staff.