The blood pressures, p mmHg, and the ages, t years, of 7 hospital patients are shown in the table below - Edexcel - A-Level Maths Statistics - Question 6 - 2010 - Paper 1
Question 6
The blood pressures, p mmHg, and the ages, t years, of 7 hospital patients are shown in the table below.
| Patient | A | B | C | D | E | F | G |
| ------- | ... show full transcript
Worked Solution & Example Answer:The blood pressures, p mmHg, and the ages, t years, of 7 hospital patients are shown in the table below - Edexcel - A-Level Maths Statistics - Question 6 - 2010 - Paper 1
Step 1
Find $S_{t}$, $S_{p}$, and $S_{tp}$ for these data.
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Answer
To find ( S_{t} ), ( S_{p} ), and ( S_{tp} ), we can use the sums given:
( S_{t} = \sum t = 341 )
( S_{p} = \sum p = 833 )
( S_{tp} = \sum tp = 42948 )
Thus, the values are:
( S_{t} = 341 )
( S_{p} = 833 )
( S_{tp} = 42948 )
Step 2
Calculate the product moment correlation coefficient for these data.
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Answer
The product moment correlation coefficient ( r ) can be calculated using the formula:
r=(nSt2−St2)(nSp2−Sp2)nStp−StSp
Where ( n = 7 ), so substituting the values we get:
Calculate ( n S_{tp} - S_{t} S_{p} )
Compute the square roots in the denominator using ( \sum t^2 ) and ( \sum p^2 ).
The final correlation coefficient will be approximately ( r = 0.70133 ).
Step 3
Interpret the correlation coefficient.
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Answer
The correlation coefficient of ( r = 0.70133 ) indicates a strong positive correlation between age and blood pressure. This suggests that as age increases, blood pressure tends to increase as well.
Step 4
On the graph paper on page 17, draw the scatter diagram of blood pressure against age for these 7 patients.
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Answer
To draw the scatter diagram, plot the points corresponding to the pairs ((t, p)) for each patient. Each point will represent the age on the x-axis and the corresponding blood pressure on the y-axis.
Step 5
Find the equation of the regression line of p on t.
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Answer
Using the least squares method, the equation of the regression line ( p = a + bt ) can be derived.
Calculate the slope ( b ) using:
b=nSt2−St2nStp−StSp
The intercept ( a ) can be calculated with:
a=nSp−bSt
This results in the equation of the regression line.
Step 6
Plot your regression line on your scatter diagram.
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Once the regression line equation is determined, it can be plotted on the scatter diagram. For this, select a few values of ( t ), calculate the corresponding ( p ) values using the regression equation, and draw the line through these points.
Step 7
Use your regression line to estimate the blood pressure of a 40 year old patient.
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Substituting ( t = 40 ) into the regression equation will provide the estimated blood pressure:
p=a+b(40)
Evaluate this expression to get the estimated blood pressure for a 40-year-old patient.
