A scientist is researching whether or not birds of prey exposed to pollutants lay eggs with thinner shells. He collects a random sample of egg shells from each of 6 different nests and tests for pollutant level, $p$, and measures the thinning of the shell, $t$. The results are shown in the table below: | $p$ | 3 | 8 | 30 | 25 | 15 | 2 | |-----|---|---|----|----|----|---| | $t$ | 1 | 3 | 9 | 10 | 5 | 6 | [You may use $\sum p^2 = 1967$ and $\sum p t = 694$] a) Draw a scatter diagram on the axes on page 7 to represent these data. b) Explain why a linear regression model may be appropriate to describe the relationship between $p$ and $t$. c) Calculate the value of $S_p$ and the value of $S_{p^2}$. d) Find the equation of the regression line of $t$ on $p$, giving your answer in the form $t = a + bp$. e) Plot the point $(\bar{p}, \bar{t})$ and draw the regression line on your scatter diagram. f) Estimate the minimum thinning of the shell that is likely to result in the death of a chick.

A-Level Edexcel Maths Statistics Working with Data: A scientist is researching wheth

A scientist is researching whether or not birds of prey exposed to pollutants lay eggs with thinner shells - Edexcel - A-Level Maths Statistics - Question 3 - 2012 - Paper 2

Question 3

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A scientist is researching whether or not birds of prey exposed to pollutants lay eggs with thinner shells. He collects a random sample of egg shells from each of 6 ... show full transcript

Worked Solution & Example Answer:A scientist is researching whether or not birds of prey exposed to pollutants lay eggs with thinner shells - Edexcel - A-Level Maths Statistics - Question 3 - 2012 - Paper 2

Step 1

a) Draw a scatter diagram

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Answer

To draw a scatter diagram, plot the points for each pair of $p$ and $t$ values on the provided axes. The $x$ -axis will represent pollutant levels ( $p$ ) and the $y$ -axis will represent shell thinning ( $t$ ). Mark each pair with a corresponding point.

Step 2

b) Explain why a linear regression model may be appropriate

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Answer

A linear regression model is appropriate because the scatter plot appears to show a linear relationship between the pollutant levels and the shell thinning. The data points seem to lie close to a straight line, indicating a correlation.

Step 3

c) Calculate the value of $S_p$ and the value of $S_{p^2}$

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The values can be calculated as follows:

The sum, $\sum p = 3 + 8 + 30 + 25 + 15 + 2 = 83$ .
For $S_p$ , we find: $S_p = \sum p = 83 \text{ and } \sum p^2 = 3^2 + 8^2 + 30^2 + 25^2 + 15^2 + 2^2 = 1967.$

Step 4

d) Find the equation of the regression line of $t$ on $p$

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Answer

The regression line is of the form $t = a + bp$ . To find $a$ and $b$ , use the formulas:

The slope, $b = \frac{S_{pt} - \frac{S_p S_t}{n}}{S_{p^2} - \frac{(S_p)^2}{n}}$ . Substitute the known values to calculate $b$ , and then compute $a$ using $a = \bar{t} - b \bar{p}$ .

Step 5

e) Plot the point $(\bar{p}, \bar{t})$

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Calculate the averages $\bar{p} = \frac{\sum p}{n}$ and $\bar{t} = \frac{\sum t}{n}$ . Plot the coordinates corresponding to these average values on the scatter diagram. Then, draw the regression line based on the previously calculated equation.

Step 6

f) Estimate the minimum thinning of the shell that is likely to result in the death of a chick

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Answer

Using the regression equation derived in part (d), substitute $p = 16$ (the pollutant level at which thinning is critical) into the equation to estimate the corresponding $t$ value. This will provide the minimum thinning of the shell likely to result in the death of a chick.

A scientist is researching whether or not birds of prey exposed to pollutants lay eggs with thinner shells - Edexcel - A-Level Maths Statistics - Question 3 - 2012 - Paper 2

Worked Solution & Example Answer:A scientist is researching whether or not birds of prey exposed to pollutants lay eggs with thinner shells - Edexcel - A-Level Maths Statistics - Question 3 - 2012 - Paper 2

a) Draw a scatter diagram

b) Explain why a linear regression model may be appropriate

c) Calculate the value of $S_p$ and the value of $S_{p^2}$

d) Find the equation of the regression line of $t$ on $p$

e) Plot the point $(\bar{p}, \bar{t})$

f) Estimate the minimum thinning of the shell that is likely to result in the death of a chick

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