(i) From the cumulative frequency diagram, determine the quartiles: \(Q_1 = 2.6\), \( median= 3.8 \text{ to } 3.85 \), \(Q_3 = 6.4 \text{ to } 6.6\). Draw a box-and-whisker plot using these values. Ensure the axis is labeled with time in seconds and is linear from 2 to 10.

(ii) Calculate the interquartile range (IQR): \(IQR = Q_3 - Q_1 = 6.5 - 2.6 = 3.9\).Calculate \(1.5 \times IQR = 1.5 \times 3.9 = 5.85\).Check for outliers: \(Q_1 - 5.85 = -3.25\) (negative, so no lower outliers) and \(Q_3 + 5.85 = 12.35\) (greater than 10, so no upper outliers).

Therefore, there are no outliers.

(i) To draw the cumulative frequency graph, calculate the cumulative frequencies:

Plot these points and join them with a smooth curve.

(ii) 28% of the daffodils are of height \(h\) cm or more, which means 72% are less than \(h\). 72% of 200 is 144, so \(h\) corresponds to the cumulative frequency of 144. From the graph, \(h \approx 22.5 \text{ cm}\).

(iii) To calculate the standard deviation:

Use the midpoints of the intervals: \(7, 13, 18, 23, 28\).

Calculate the variance:

\(\text{var} = \frac{(7^2 \times 22 + 13^2 \times 32 + 18^2 \times 78 + 23^2 \times 40 + 28^2 \times 28)}{200} - 18.39^2\)

\(= \frac{74870}{200} - 18.39^2\)

\(= 374.35 - 18.39^2\)

\(= 36.1579\)

Standard deviation \(\text{sd} = \sqrt{36.1579} = 6.01\).

(i) Plot the cumulative frequency points: (40, 0), (50, 12), (60, 34), (65, 64), (70, 92), (90, 144). Draw a smooth curve through these points.

(ii) To find \(c\), note that 64 people weigh more than \(c\), so 80 people weigh less than \(c\). From the graph, this corresponds to a weight of 67.2 kg. Thus, \(c = 67.2\) kg.

(iii) Calculate the frequencies: 12, 22, 30, 28, 52. Use midpoints: 45, 55, 62.5, 67.5, 80. Calculate the mean:

\(\text{Mean} = \frac{(45 \times 12) + (55 \times 22) + (62.5 \times 30) + (67.5 \times 28) + (80 \times 52)}{144} = \frac{9675}{144} = 67.2 \text{ kg}\)

\(Calculate the variance:\)

\(\text{Variance} = \frac{(45^2 \times 12) + (55^2 \times 22) + (62.5^2 \times 30) + (67.5^2 \times 28) + (80^2 \times 52)}{144} - (67.2)^2 = 127.59\)

Standard deviation:

\(\text{Standard deviation} = \sqrt{127.59} = 11.3 \text{ kg}\)

(i) From the cumulative frequency graph, determine the lower quartile (LQ), median, and upper quartile (UQ):

• LQ = 15 (thousands of euros)
• Median = 18 (thousands of euros)
• UQ = 26 (thousands of euros)

Draw a box-and-whisker plot with these values, using a linear scale from 0 to 80 (thousands of euros). The whiskers extend from the minimum (5) to LQ and from UQ to the maximum (80).

(ii) Comment: Most (3/4) of the sample earn less than 26,000 euros, indicating not many earn high salaries.

(iii) Calculate the interquartile range (IQR):

\(\text{IQR} = UQ - LQ = 26 - 15 = 11\)

(a) A high outlier is any salary above:

\(UQ + 1.5 \times \text{IQR} = 26 + 1.5 \times 11 = 42.5 \text{ (thousands of euros)} = 42500 \text{ euros}\)

(b) A low outlier is any salary below:

\(LQ - 1.5 \times \text{IQR} = 15 - 1.5 \times 11 = -1.5 \text{ (thousands of euros)}\)

Since the lowest salary is 5000 euros, no salaries are low enough to be classified as outliers.