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 these sums, we first need to determine the total values:

• The total of t: $\Sigma t = 212$

• The total of Em: $\Sigma Em = 61$

• The product of t and Em: $\Sigma tm = 2485$

• Calculation of $S_t$ using: $S_t = \frac{\Sigma Em}{n} = \frac{2485}{10} = 248.5$

• Calculation of $S_{Em}$ using: $S_{Em} = \frac{\Sigma Em}{n} = \frac{61}{10} = 6.1$

The product moment correlation coefficient can be calculated using the formula:

$r = \frac{n\Sigma tm - \Sigma t \Sigma Em}{\sqrt{(n \Sigma t^2 - (\Sigma t)^2)(n \Sigma m^2 - (\Sigma m)^2)}}$

Plugging in the values:

$n = 10,$
$\Sigma t = 212,$
$\Sigma Em = 61,$
$\Sigma tm = 2485,$
$\Sigma t^2 = 5478,$
$\Sigma Em^2 = 2101$

Now, substituting:

$r = \frac{10 \times 2485 - 212 \times 61}{\sqrt{(10 \times 5478 - 212^2)(10 \times 2101 - 61^2)}}$

Calculating the numerator and denominator to find:

$r \approx 0.914$

The product moment correlation coefficient between $t$ and the actual amount spent is 0.914. This indicates a strong positive correlation, suggesting that as the time spent increases, the amount spent also tends to increase.

The correlation coefficient of 0.178 from the similar data suggests a weaker relationship between time spent and actual amount spent compared to 0.914. This may imply that on different days or contexts, factors such as day of the week or customer behavior affect spending patterns.

A practical reason for the difference in the correlation coefficients could be variations in shopper behavior on different days, such as different customer demographics or shopping habits, which can influence the amount spent.