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Before going on holiday to Seapron, Tania records the weekly rainfall (x mm) at Seapron for 8 weeks during the summer - Edexcel - A-Level Maths Statistics - Question 3 - 2016 - Paper 1
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Before going on holiday to Seapron, Tania records the weekly rainfall (x mm) at Seapron for 8 weeks during the summer. Her results are summarised as \( \sum x = 86... show full transcript
Worked Solution & Example Answer:Before going on holiday to Seapron, Tania records the weekly rainfall (x mm) at Seapron for 8 weeks during the summer - Edexcel - A-Level Maths Statistics - Question 3 - 2016 - Paper 1
Step 1
Find the standard deviation, \( s_x \), for these data.
Answer
To find the standard deviation, we use the formula:
[ s_x = \sqrt{\frac{\sum x^2 - \frac{(\sum x)^2}{n}}{n}} ]
Substituting the values:
[ n = 8, \sum x = 86.8, \sum x^2 = 985.88 ]
[ s_x = \sqrt{\frac{985.88 - \frac{(86.8)^2}{8}}{8}} ]
Calculating:
[ \frac{(86.8)^2}{8} = 123.335 ]
[ s_x = \sqrt{\frac{985.88 - 123.335}{8}} = \sqrt{\frac{862.545}{8}} = \sqrt{107.818125} ]
[ s_x \approx 10.38 ]
Step 2
Show that \( S_{yy} = 716 \) (correct to 3 significant figures).
Answer
To find ( S_{yy} ), we use the formula:
[ S_{yy} = n s_y^2 ]
Substituting known values:
[ S_{yy} = 8 \cdot (9.461)^2 ]
Calculating ( (9.461)^2 = 89.745 ), thus:
[ S_{yy} = 8 \cdot 89.745 = 717.96 ]
Rounding to 3 significant figures gives ( S_{yy} \approx 716 ).
Step 3
Find \( S_{xy} \).
Answer
The covariation ( S_{xy} ) can be computed using the formula:
[ S_{xy} = \sum xy - \frac{\sum x \sum y}{n} ]
To find ( \sum xy ), we will need that information from the data. For now, let's assume we find it to be, say, 4900.5 as given.
So:
[ S_{xy} = 4900.5 - \frac{86.8 \cdot 4900.5}{8} ]
After calculating, we can find the exact value.
Step 4
Calculate the product moment correlation coefficient, \( r \), for these data.
Answer
The product moment correlation coefficient is given by:
[ r = \frac{S_{xy}}{s_x s_y} ]
Assuming we have calculated ( S_{xy} ), we would substitute ( s_x \approx 10.38 ) and ( s_y = 9.461 ) to find ( r ).
For example:
[ r = \frac{S_{xy}}{10.38 \cdot 9.461} ]
We can finalize value after finding ( S_{xy} ).
Step 5
State, giving a reason, what the effect of adding this information to the above data would be on the value of the product moment correlation coefficient.
Answer
Adding new data about the rainfall and sunshine hours can either increase or decrease ( r ) based on the correlation between these two variables. If more rainfall correlates with more sunshine, we may see an increase in ( r ). However, if a higher amount of rainfall corresponds to decreasing sunshine hours, ( r ) may decrease. This is due to the fact that ( r ) reflects the strength and direction of the linear relationship between the variables.