The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015. An outlier is defined as a value more than 1.5 × IQR below $Q_1$, or more than 1.5 × IQR above $Q_3$. The three lowest air temperatures in the data set are 7.6°C, 8.1°C and 9.1°C. The highest air temperature in the data set is 32.5°C. a) Complete the box plot in Figure 1 showing clearly any outliers. b) Using your knowledge of the large data set, suggest from which month the two outliers are likely to have come. Using the data from the large data set, Simon produced the following summary statistics for the daily mean air temperature, $x^C$, for Beijing in 2015: $n = 184$ $ar{x} = 4153.6$ $S_{x}^{2} = 4952.906$ c) Show that, to 3 significant figures, the standard deviation is 5.19°C. Simon decides to model the air temperatures with the random variable $T ∼ N(22, 6.519)$. d) Using Simon’s model, calculate the 10th to 90th interpercentile range. e) Simon wants to model another variable from the large data set for Beijing using a normal distribution. State two variables from the large data set for Beijing that are not suitable to be modeled by a normal distribution. Give a reason for each answer.

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The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015 - Edexcel - A-Level Maths Statistics - Question 2 - 2019 - Paper 1

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The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015. An ou... show full transcript

Worked Solution & Example Answer:The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015 - Edexcel - A-Level Maths Statistics - Question 2 - 2019 - Paper 1

Step 1

Complete the box plot in Figure 1 showing clearly any outliers.

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Answer

To identify any outliers based on the provided definition, we first need to calculate the Interquartile Range (IQR). The IQR is calculated as follows:

$IQR = Q_3 - Q_1$

Given that $Q_1 = 8.1°C$ and $Q_3 = 26.6°C$ (from the data), we find:

$IQR = 26.6 - 8.1 = 18.5°C$

Next, we calculate 1.5 times the IQR:

$1.5 imes IQR = 1.5 imes 18.5 = 27.75°C$

Now, we determine the lower and upper bounds for outliers:

Lower Bound: $Q_1 - 1.5 imes IQR = 8.1 - 27.75 = -19.65°C$
Upper Bound: $Q_3 + 1.5 imes IQR = 26.6 + 27.75 = 54.35°C$

After analyzing the data, the values of temperature below -19.65°C or above 54.35°C can be considered outliers. In the context of the provided data, the significant outlier is the highest temperature, which is 32.5°C. We plot this alongside the calculated quartiles and whiskers to complete the box plot.

Step 2

Using your knowledge of the large data set, suggest from which month the two outliers are likely to have come.

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Answer

The significant outlier temperatures are typically expected during extreme weather conditions. Given the temperatures provided for Beijing and historical weather patterns, we can suggest that the outlier temperatures likely come from the months of October or January, as these are historically colder months in the region.

Step 3

Show that, to 3 significant figures, the standard deviation is 5.19°C.

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Answer

To find the standard deviation, we use the relationship:

$S_x = rac{ au_s}{ au}$

where $S_{x}^{2} = 4952.906$ and $n = 184$ . The variance $S_{x}^{2}$ can be expressed as:

$S_x = ext{sqrt}(S_{x}^{2}) = ext{sqrt}(4952.906)$

Calculating this gives:

$S_x ≈ 70.4$

Now adjusting for standard deviation based on the size of the sample:

$S_x ≈ rac{70.4}{ ext{sqrt}(n)} = rac{70.4}{ ext{sqrt}(184)}$

Finally, calculating gives us the value close to 5.19°C when rounded to three significant figures.

Step 4

Using Simon’s model, calculate the 10th to 90th interpercentile range.

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Answer

To calculate the 10th to 90th interpercentile range for $T ∼ N(22, 6.519)$ , we find the corresponding z-scores for the 10th and 90th percentiles:

For the 10th percentile: $z_{0.10} ≈ -1.2816$
For the 90th percentile: $z_{0.90} ≈ 1.2816$

Using the z-score formula: $X = ext{mean} + (z imes ext{standard deviation})$

Calculating these values gives:

At the 10th percentile: $X_{10} = 22 + (-1.2816 imes 6.519) ≈ 15.94°C$
At the 90th percentile: $X_{90} = 22 + (1.2816 imes 6.519) ≈ 29.25°C$

Thus, the interpercentile range is: $ext{IPR} = X_{90} - X_{10} ≈ 29.25 - 15.94 ≈ 13.31°C$

Step 5

State two variables from the large data set for Beijing that are not suitable to be modeled by a normal distribution. Give a reason for each answer.

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Answer

Rainfall: Rainfall data is often skewed with many days receiving no rain at all, leading to a non-normal distribution.
Daily wind speed: Wind speeds can be affected by extreme weather events and are typically not symmetrically distributed, which deviates from the assumption of normality.

The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015 - Edexcel - A-Level Maths Statistics - Question 2 - 2019 - Paper 1

Worked Solution & Example Answer:The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015 - Edexcel - A-Level Maths Statistics - Question 2 - 2019 - Paper 1

Complete the box plot in Figure 1 showing clearly any outliers.

Using your knowledge of the large data set, suggest from which month the two outliers are likely to have come.

Show that, to 3 significant figures, the standard deviation is 5.19°C.

Using Simon’s model, calculate the 10th to 90th interpercentile range.

State two variables from the large data set for Beijing that are not suitable to be modeled by a normal distribution. Give a reason for each answer.

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