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A university uses gas to heat its buildings - HSC - SSCE Mathematics Advanced - Question 18 - 2023 - Paper 1

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A university uses gas to heat its buildings. Over a period of 10 weekdays during winter, the gas used each day was measured in megawatts (MW) and the average outside... show full transcript

Worked Solution & Example Answer:A university uses gas to heat its buildings - HSC - SSCE Mathematics Advanced - Question 18 - 2023 - Paper 1

Step 1

Calculate the means $ar{x}$ and $ar{y}$

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Answer

The average temperature, ar{x}, is calculated as follows:

ar{x} = \frac{0 + 0 + 2 + 5 + 7 + 8 + 9 + 9 + 10}{10} = 5

The average gas usage, ar{y}, is:

ar{y} = \frac{1840}{10} = 184

Thus, the point (xˉ,yˉ)(\bar{x}, \bar{y}) is (5,184)(5, 184).

Step 2

Find the equation of the regression line

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Answer

To find the slope of the regression line mm, we use the following formula:

extslope=yˉ−236xˉ−0=184−2365−0=−10.4 ext{slope} = \frac{\bar{y} - 236}{\bar{x} - 0} = \frac{184 - 236}{5 - 0} = -10.4

This gives us the equation of the regression line as:

y = 236 - 10.4x.

Step 3

Identify ONE problem with predicting gas usage

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Answer

When the temperature is 23°C, the regression equation predicts a negative gas usage:

y=236−10.4(23)y = 236 - 10.4(23)

This results in:

y = 236 - 239.2 = -3.2 MW,

which is not physically possible, as gas usage cannot be negative.

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