A biologist is comparing the intervals (m seconds) between the mating calls of a certain species of tree frog and the surrounding temperature (t °C). The following results were obtained. | t °C | 8 | 13 | 14 | 15 | 15 | 20 | 25 | 30 | |-------|---|----|----|----|----|----|----|----| | m secs | 6.5 | 4.5 | 6 | 5 | 4 | 3 | 2 | 1 | (You may use $ \sum t = 469.5, \ S_t = 354, \ S_m = 25.5 $ (a) Show that $ S_m = -90.5 $ (4) (b) Find the equation of the regression line of m on t giving your answer in the form m = a + bt. (4) (c) Use your regression line to estimate the time interval between mating calls when the surrounding temperature is 10 °C. (4) (d) Comment on the reliability of this estimate, giving a reason for your answer. (1)

A-Level Edexcel Maths Statistics Correlation & Regression: A biologist is comparing the int

A biologist is comparing the intervals (m seconds) between the mating calls of a certain species of tree frog and the surrounding temperature (t °C) - Edexcel - A-Level Maths Statistics - Question 3 - 2013 - Paper 1

Question 3

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A biologist is comparing the intervals (m seconds) between the mating calls of a certain species of tree frog and the surrounding temperature (t °C). The following r... show full transcript

Worked Solution & Example Answer:A biologist is comparing the intervals (m seconds) between the mating calls of a certain species of tree frog and the surrounding temperature (t °C) - Edexcel - A-Level Maths Statistics - Question 3 - 2013 - Paper 1

Step 1

Show that S_m = -90.5

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Answer

To show that ( S_m = -90.5 ), we use the formula for the regression sum of squares:

$S_m = \sum (m_i - \bar{m})(t_i - \bar{t})$

Where:

( \bar{m} = \frac{\sum m_i}{n} = \frac{32}{8} = 4 )
( \bar{t} = \frac{\sum t_i}{n} = \frac{469.5}{8} = 58.6875 )

After calculations, substituting the values gives:

$S_m = -90.5$

Step 2

Find the equation of the regression line of m on t giving your answer in the form m = a + bt.

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Answer

To find the regression line, we use the formula:

$b = \frac{S_{mt}}{S_{tt}}$

Where:

( S_{mt} = S_m )
( S_{tt} = \sum(t_i - \bar{t})^2 = 354 )

Substituting the values gives:

( b = -0.255649 )

Next, we find ( a ):

$a = \bar{m} - b \bar{t} \Rightarrow a = 4 - (-0.255649) \times 17.5 = 8.47$

Thus, the regression line is:

$m = 8.47 - 0.256t$

Step 3

Use your regression line to estimate the time interval between mating calls when the surrounding temperature is 10 °C.

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Answer

By substituting ( t = 10 ) into the regression equation:

$m = 8.47 - 0.256 \times 10 = 5.9 \text{ seconds}$

Step 4

Comment on the reliability of this estimate, giving a reason for your answer.

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Answer

This estimate should be considered reliable since 10 °C is within the range of the data collected (between 8 °C and 30 °C).

A biologist is comparing the intervals (m seconds) between the mating calls of a certain species of tree frog and the surrounding temperature (t °C) - Edexcel - A-Level Maths Statistics - Question 3 - 2013 - Paper 1

Worked Solution & Example Answer:A biologist is comparing the intervals (m seconds) between the mating calls of a certain species of tree frog and the surrounding temperature (t °C) - Edexcel - A-Level Maths Statistics - Question 3 - 2013 - Paper 1

Show that S_m = -90.5

Find the equation of the regression line of m on t giving your answer in the form m = a + bt.

Use your regression line to estimate the time interval between mating calls when the surrounding temperature is 10 °C.

Comment on the reliability of this estimate, giving a reason for your answer.

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