Each member of a group of 27 people was timed when completing a puzzle. The time taken, x minutes, for each member of the group was recorded. These times are summarised in the following box and whisker plot. (a) Find the range of the times. (b) Find the interquartile range of the times. For these 27 people $\, \sum x = 607.5 \,$ and $\, \sum x^2 = 17623.25 \,$ (c) calculate the mean time taken to complete the puzzle. (d) calculate the standard deviation of the times taken to complete the puzzle. Taruni defines an outlier as a value more than 3 standard deviations above the mean. (e) State how many outliers Taruni would say there are in these data, giving a reason for your answer. (f) Adam and Beth also completed the puzzle in a minutes and b minutes respectively, where a > b. When their times are included with the data of the other 27 people - the median time increases - the mean time does not change. (g) Suggest a possible value for a and a possible value for b, explaining how your values satisfy the above conditions. (h) Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).

A-Level Edexcel Maths Statistics Hypothesis Testing: Each member of a group of 27 peo

Each member of a group of 27 people was timed when completing a puzzle - Edexcel - A-Level Maths Statistics - Question 3 - 2020 - Paper 1

Question 3

Each member of a group of 27 people was timed when completing a puzzle. The time taken, x minutes, for each member of the group was recorded. These times are summa... show full transcript

Worked Solution & Example Answer:Each member of a group of 27 people was timed when completing a puzzle - Edexcel - A-Level Maths Statistics - Question 3 - 2020 - Paper 1

Step 1

Find the range of the times.

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Answer

To find the range, we subtract the minimum time from the maximum time. From the box and whisker plot:

Maximum time = 68 minutes
Minimum time = 7 minutes

Thus, the range is calculated as follows:

Range = Maximum - Minimum = 68 - 7 = 61 minutes.

Step 2

Find the interquartile range of the times.

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Answer

The interquartile range (IQR) is determined by subtracting the first quartile (Q1) from the third quartile (Q3). From the box plot:

Q1 = 14 minutes
Q3 = 25 minutes

Therefore, IQR = Q3 - Q1 = 25 - 14 = 11 minutes.

Step 3

calculate the mean time taken to complete the puzzle.

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The mean time can be calculated using the formula:

$\mu = \frac{\sum x}{n}$ where ( n ) is the number of observations.

Given that $\sum x = 607.5$ and there are 27 people:

$\mu = \frac{607.5}{27} = 22.5 \text{ minutes}$ .

Step 4

calculate the standard deviation of the times taken to complete the puzzle.

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Answer

The standard deviation (σ) can be calculated using:

$\sigma = \sqrt{\frac{\sum x^2 - \frac{(\sum x)^2}{n}}{n}}$

Substituting the known values:

$\sigma = \sqrt{\frac{17623.25 - \frac{(607.5)^2}{27}}{27}}$ Calculating this gives: $\sigma = \sqrt{\frac{17623.25 - 14061.25}{27}} = \sqrt{\frac{3556}{27}} \approx 12.1 \text{ minutes}.$

Step 5

State how many outliers Taruni would say there are in these data, giving a reason for your answer.

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Answer

First calculate the mean (μ) and standard deviation (σ):

Mean (μ) = 22.5
Standard deviation (σ) = 12.1

An outlier is defined as a data point that is greater than: $\mu + 3\sigma$

Calculating: $22.5 + 3(12.1) = 22.5 + 36.3 = 58.8$

Any time greater than 58.8 is considered an outlier. From the data, only one time (the maximum time = 68 minutes) exceeds this value. Therefore, Taruni would say there is 1 outlier.

Step 6

Suggest a possible value for a and a possible value for b, explaining how your values satisfy the above conditions.

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Answer

To maintain the scenario where the median time increases but the mean remains unchanged, we select:

a = 45 (greater than b)
b = 40 (less than 45)

In this case:

Adding b=40 reduces the median slightly.
Adding a=45 keeps the average (mean) the same or slightly increases it while keeping the median in check, thus satisfying both conditions.

Step 7

Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).

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Answer

When adding two additional times (from Adam and Beth), if these values (40 and 45) are both less than 1 standard deviation away from the mean (22.5), they are effectively bringing the overall distribution closer together. Since these values are closer to the mean than many of the existing data points, the variability of the overall dataset is reduced, leading to a lower standard deviation.

Each member of a group of 27 people was timed when completing a puzzle - Edexcel - A-Level Maths Statistics - Question 3 - 2020 - Paper 1

Worked Solution & Example Answer:Each member of a group of 27 people was timed when completing a puzzle - Edexcel - A-Level Maths Statistics - Question 3 - 2020 - Paper 1

Find the range of the times.

Find the interquartile range of the times.

calculate the mean time taken to complete the puzzle.

calculate the standard deviation of the times taken to complete the puzzle.

State how many outliers Taruni would say there are in these data, giving a reason for your answer.

Suggest a possible value for a and a possible value for b, explaining how your values satisfy the above conditions.

Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).

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