A clothes shop manager records the weekly sales figures, £ s, and the average weekly temperature, t °C, for 6 weeks during the summer - Edexcel - A-Level Maths Statistics - Question 1 - 2017 - Paper 1

From the information provided:

[ S_{xx} = \sum t^2 - \frac{(\sum t)^2}{n} ] Substituting the values: [ S_{xx} = 2435 - \frac{119^2}{6} ] [ S_{xx} = 2435 - \frac{14161}{6} = 2435 - 2360.1667 = 74.8333 ] Thus, ( S_{xx} \approx 74.8 ), and ( S_{t} ) remains ( 119 ).

The product moment correlation coefficient ( r ) can be calculated using:

[ r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}} ] Where ( S_{xy} = \sum w t - \frac{(\sum w)(\sum t)}{n} ) and ( n = 6 ). Substituting the values: [ S_{xy} = 784 - \frac{42 \cdot 119}{6} = 784 - 833 = -49 ] Therefore, [ r = \frac{-49}{\sqrt{74.8333 \cdot S_{yy}}} ] (You would need to calculate ( S_{yy} ) separately based on given data.)

The computed correlation coefficient ( r ) is a negative value, implying that as temperature increases, sales decrease. This shows a negative relationship, indicating that it does not support the manager's belief of a positive linear regression.

The equation of the regression line is given by:

[ w = a + bt ] Where: [ b = \frac{S_{xy}}{S_{xx}} ] From previous calculations, substituting the corresponding values: [ b = \frac{-49}{74.8333} \approx -0.6547 ] Next, substituting the values to find ( a ): [ a = \bar{w} - b \bar{t} ] Calculate ( \bar{w} ) and ( \bar{t} ) to find ( a ) and finalize the equation.

Using the relationship between sales and coded sales, we can rewrite it as:

[ s = 1000(w) = 1000(a + bt) = 1000a + 1000bt ] Where ( c = 1000a ) and ( d = 1000b ). Substituting the found values of ( a ) and ( b ): [ s = c + dt ] with ( c ) and ( d ) calculated to 3 significant figures.

The value for ( d ) from part (f) indicates how much sales change with an increase in temperature. If ( d = -655 ), this suggests that for each additional degree Celsius increase in temperature, weekly sales would decrease by £ 655. This negative effect reflects a direct inverse relationship between temperature and sales.