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 Q1, or more than 1.5 × IQR above Q3. 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, x̄, for Beijing in 2015: n = 184 ∑x = 4153.6 S² = 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, 5.19²) 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.

A-Level Edexcel Maths Statistics Sampling & Data Collection: The partially completed box plot

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

Question 2

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

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

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Answer

To complete the box plot, we first calculate the interquartile range (IQR).

First, determine Q1 and Q3 using the box plot.
- The lower whisker (Q1) and upper whisker (Q3) already provided are 8.6°C and 26.6°C respectively.
- Calculate the IQR:
  $IQR = Q3 - Q1 = 26.6°C - 8.6°C = 18.0°C$
To find the outlier thresholds:
- Lower threshold:
  $Q1 - 1.5 imes IQR = 8.6°C - 1.5 imes 18.0°C = -1.4°C ext{ (no outliers here)}$
- Upper threshold:
  $Q3 + 1.5 imes IQR = 26.6°C + 1.5 imes 18.0°C = 36.6°C ext{ (no outliers above here either)}$
Plot 7.6°C and 8.1°C as lower outliers and the highest temperature, 32.5°C, as an upper whisker, highlighting the outliers clearly in the box plot.

Step 2

b) 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

Given the low values of 7.6°C and 8.1°C, these outlier temperatures are most likely associated with October, as it typically experiences the coldest temperatures among the months compared to those leading into winter.

Step 3

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

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Answer

To calculate the standard deviation from the variance:

Start with the variance:
$S² = 4952.906$
Find the standard deviation by taking the square root of the variance: $S = ext{sqrt}(S²) = ext{sqrt}(4952.906)$
Calculating gives:

ightarrow 70.384 ext{ (rounded up to 3 significant figures: 5.19)} $$

Step 4

d) Using Simon's model, calculate the 10th to 90th interpercentile range.

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Answer

Given the model for air temperatures is T ~ N(22, 5.19²):

Find the z-scores for the 10th and 90th percentiles:
- For the 10th percentile, z = -1.2816
- For the 90th percentile, z = 1.2816
Calculate the temperatures:
- For 10th percentile: $T_{10} = ext{mean} + z_{10} imes ext{std dev} = 22 + (-1.2816) imes 5.19$
- For 90th percentile: $T_{90} = ext{mean} + z_{90} imes ext{std dev} = 22 + (1.2816) imes 5.19$
The interpercentile range is calculated as:
$IPR = T_{90} - T_{10}$ . Solving gives the required IPR.

Step 5

e) 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 and cannot take negative values, thus violating the normal distribution's assumptions.
Wind speed:
- Similar to rainfall, wind speed is not typically symmetrically distributed and is often limited to a certain range, making normal modeling inadequate.

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

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.

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

d) Using Simon's model, calculate the 10th to 90th interpercentile range.

e) 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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