The problem provides a table with the masses of parcels in different ranges and their corresponding frequencies. Part (i) asks to complete the histogram. Part (ii) asks to estimate the mean mass of the parcels. I'll address both parts. The histogram requires calculating frequency densities. The mean mass calculation requires using the midpoints of the mass intervals.

1. Problem Description

The problem provides a table with the masses of parcels in different ranges and their corresponding frequencies. Part (i) asks to complete the histogram. Part (ii) asks to estimate the mean mass of the parcels. I'll address both parts. The histogram requires calculating frequency densities. The mean mass calculation requires using the midpoints of the mass intervals.

2. Solution Steps

(i) Completing the Histogram:

First, we need to calculate the frequency density for each mass interval. The frequency density is calculated by dividing the frequency by the class width.

Frequency Density = Frequency / Class Width

*

0. 5 < m ≤ 1: Class Width = 1 - 0.5 = 0.

5. Frequency Density = 4 / 0.5 =

8. The bar is already drawn correctly to height

8. * 1 < m ≤ 2: Class Width = 2 - 1 =

1. Frequency Density = 7 / 1 =

7. The bar is already drawn correctly to height

7. * 2 < m ≤ 4: Class Width = 4 - 2 =

2. Frequency Density = 15 / 2 = 7.

5. We need to draw a bar from 2 to 4 with a height of 7.

5. * 4 < m ≤ 7: Class Width = 7 - 4 =

3. Frequency Density = 10 / 3 = 3.33 (approximately). We need to draw a bar from 4 to 7 with a height of 3.

3

3. * 7 < m ≤ 12: Class Width = 12 - 7 =

5. Frequency Density = 4 / 5 = 0.

8. We need to draw a bar from 7 to 12 with a height of 0.

8.

(ii) Calculating the Estimated Mean Mass:

To calculate the estimated mean mass, we use the midpoint of each interval multiplied by its frequency, then sum these values and divide by the total frequency (40).

Midpoint = (Lower Bound + Upper Bound) / 2

*

0. 5 < m ≤ 1: Midpoint = (0.5 + 1) / 2 = 0.

7

5. Frequency =

4. * 1 < m ≤ 2: Midpoint = (1 + 2) / 2 = 1.

5. Frequency =

7. * 2 < m ≤ 4: Midpoint = (2 + 4) / 2 =

3. Frequency =

1

5. * 4 < m ≤ 7: Midpoint = (4 + 7) / 2 = 5.

5. Frequency =

1

0. * 7 < m ≤ 12: Midpoint = (7 + 12) / 2 = 9.

5. Frequency =

4.

Sum of (Midpoint * Frequency):

$(0.75 * 4) + (1.5 * 7) + (3 * 15) + (5.5 * 10) + (9.5 * 4) = 3 + 10.5 + 45 + 55 + 38 = 151.5$

Estimated Mean Mass = Sum of (Midpoint * Frequency) / Total Frequency

Estimated Mean Mass = 151.5 / 40 = 3.7875

3. Final Answer

(i) The histogram has been completed with bars as follows:

* 2 < m ≤ 4: Height = 7.5

* 4 < m ≤ 7: Height = 3.33

* 7 < m ≤ 12: Height = 0.8

(ii) The estimated mean mass of the parcels is 3.7875 kg.