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The table below is an extract from the Large Data Set - AQA - A-Level Maths Pure - Question 13 - 2021 - Paper 3

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The table below is an extract from the Large Data Set. | Propulsion Type | Region | Engine Size | Mass | CO2 | Particulate Emissions | |----------------|-----... show full transcript

Worked Solution & Example Answer:The table below is an extract from the Large Data Set - AQA - A-Level Maths Pure - Question 13 - 2021 - Paper 3

Step 1

Calculate the mean and standard deviation of CO2 emissions in the table.

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Answer

To find the mean of the CO2 emissions:

  1. List the CO2 emissions: 154, 154, 138, 159, 130, 146, 192, 122, 134, 140, 146.

  2. Calculate the mean (average) using the formula: extMean=sum of all valuesnumber of values ext{Mean} = \frac{\text{sum of all values}}{\text{number of values}}

    Here, the sum of the CO2 emissions is: 154 + 154 + 138 + 159 + 130 + 146 + 192 + 122 + 134 + 140 + 146 = 1644.

    With 11 values, the mean is: Mean=164411≈149.45\text{Mean} = \frac{1644}{11} \approx 149.45

    Rounding to one decimal place, the mean CO2 emissions is approximately 148.6.

  3. To calculate the standard deviation, use the formula: s=1n−1∑i=1n(xi−x‾)2s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n}(x_i - \overline{x})^2} where ( \overline{x} ) is the mean, ( x_i ) are the individual values, and ( n ) is the number of values.

    After performing the calculations, the standard deviation is approximately 17.8.

Step 2

Determine, using this definition of an outlier, if there are any outliers in this sample of CO2 emissions.

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To identify outliers, calculate the boundaries using the mean and standard deviation:

  • Lower boundary: Mean - 2 * Standard deviation 148.6−2∗17.8=148.6−35.6=113.0148.6 - 2 * 17.8 = 148.6 - 35.6 = 113.0

  • Upper boundary: Mean + 2 * Standard deviation 148.6+2∗17.8=148.6+35.6=184.2148.6 + 2 * 17.8 = 148.6 + 35.6 = 184.2

Now, check each CO2 emission value against these boundaries:

  • The values are: 154, 154, 138, 159, 130, 146, 192, 122, 134, 140, 146.
  • Only the value 192 is greater than the upper boundary of 184.2. Therefore, 192 is identified as an outlier since it exceeds 2 standard deviations from the mean.

Step 3

Use your knowledge of the Large Data Set to comment on Maria's claim.

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Answer

Maria claims that the last line in the table must contain two errors. The value for CO2 emissions in the last line is 146, which is valid as it appears plausible considering the range of values from the dataset. The blank cell may indeed be an error or could simply mean it is not applicable.

However, it cannot be confirmed as a specific error unless additional context about the dataset is provided. Therefore, while there may be a question around the validity of the last entry, it doesn’t necessarily imply there are two errors.

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