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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) - Edexcel - A-Level Maths Statistics - Question 3 - 2013 - Paper 1

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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:

Sm=(mimˉ)(titˉ)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:

Sm=90.5S_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=SmtSttb = \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=mˉbtˉa=4(0.255649)×17.5=8.47a = \bar{m} - b \bar{t} \Rightarrow a = 4 - (-0.255649) \times 17.5 = 8.47

Thus, the regression line is:

m=8.470.256tm = 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=8.470.256×10=5.9 secondsm = 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).

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