The problem provides a dataset of average phone call durations (in minutes). We need to: (i) Convert the data into a grouped frequency table using Sturges' rule. (ii) Plot the less than and more than cumulative frequency curves on the same graph. (iii) Compute the geometric mean. (iv) Compute the median. (v) Compute the mode of the distribution.

1. Problem Description

The problem provides a dataset of average phone call durations (in minutes). We need to:

(i) Convert the data into a grouped frequency table using Sturges' rule.

(ii) Plot the less than and more than cumulative frequency curves on the same graph.

(iii) Compute the geometric mean.

(iv) Compute the median.

(v) Compute the mode of the distribution.

2. Solution Steps

(i) Converting to a Grouped Frequency Table using Sturges' Rule:

First, determine the number of classes (k) using Sturges' rule:

$k = 1 + 3.322 * log_{10}(n)$

Where n is the number of observations.

In this case, we have a $6 \times 10$ table, so $n = 60$.

$k = 1 + 3.322 * log_{10}(60) = 1 + 3.322 * 1.778 \approx 1 + 5.906 \approx 6.906$

We round this to the nearest whole number, so $k = 7$.

Next, we find the range of the data:

Range = Maximum value - Minimum value = $135 - 12 = 123$

Now, we calculate the class width (w):

$w = \frac{Range}{k} = \frac{123}{7} \approx 17.57$

We can round this up to 18 for convenience.

Now we can construct the grouped frequency table. Start with the minimum value and create 7 classes of width

1

8.

| Class Interval | Frequency (f) |

|---|---|

| 12-29 | 10 |

| 30-47 | 9 |

| 48-65 | 11 |

| 66-83 | 10 |

| 84-101 | 6 |

| 102-119 | 8 |

| 120-137 | 6 |

(ii) Plotting Cumulative Frequency Curves:

To plot the cumulative frequency curves, we need to calculate the less than and more than cumulative frequencies.

Less than Cumulative Frequency:

| Upper Class Boundary | Less Than Cumulative Frequency |

|---|---|

| 29 | 10 |

| 47 | 19 |

| 65 | 30 |

| 83 | 40 |

| 101 | 46 |

| 119 | 54 |

| 137 | 60 |

More than Cumulative Frequency:

| Lower Class Boundary | More Than Cumulative Frequency |

|---|---|

| 12 | 60 |

| 30 | 50 |

| 48 | 41 |

| 66 | 30 |

| 84 | 20 |

| 102 | 14 |

| 120 | 6 |

Plot these two curves on a graph, with the x-axis representing the class boundaries and the y-axis representing the cumulative frequency.

(iii) Computing the Geometric Mean (GM):

Since we have grouped data, we approximate each value as the class midpoint.

| Class Interval | Midpoint (x) | Frequency (f) | f*log(x) |

|---|---|---|---|

| 12-29 | 20.5 | 10 | 13.118 |

| 30-47 | 38.5 | 9 | 14.119 |

| 48-65 | 56.5 | 11 | 19.220 |

| 66-83 | 74.5 | 10 | 18.722 |

| 84-101 | 92.5 | 6 | 11.626 |

| 102-119 | 110.5 | 8 | 16.624 |

| 120-137 | 128.5 | 6 | 12.909 |

| | | Sum(f) = 60 | Sum(f*log(x)) = 106.338 |

$log(GM) = \frac{\sum f * log(x)}{\sum f}$

$log(GM) = \frac{106.338}{60} \approx 1.7723$

$GM = 10^{1.7723} \approx 59.19$

(iv) Computing the Median:

Median is the middle value. Since $n = 60$, the median lies between the 30th and 31st values. The 30th data falls inside the class interval 48-

6

5.

$Median = L + (\frac{\frac{n}{2} - cf}{f}) * w$

L = Lower boundary of the median class = 47.5

n = Total frequency = 60

cf = Cumulative frequency of the class preceding the median class = 19

f = Frequency of the median class = 11

w = Class width = 18

$Median = 47.5 + (\frac{30 - 19}{11}) * 18$

$Median = 47.5 + (\frac{11}{11}) * 18 = 47.5 + 18 = 65.5$

(v) Computing the Mode:

Mode is the value that appears most frequently. From the grouped frequency table, the class interval 48-65 has the highest frequency (11).

$Mode = L + (\frac{f_m - f_1}{2f_m - f_1 - f_2}) * w$

L = Lower boundary of the modal class = 47.5

$f_m$ = Frequency of the modal class = 11

$f_1$ = Frequency of the class preceding the modal class = 9

$f_2$ = Frequency of the class following the modal class = 10

w = Class width = 18

$Mode = 47.5 + (\frac{11 - 9}{2*11 - 9 - 10}) * 18$

$Mode = 47.5 + (\frac{2}{22 - 19}) * 18 = 47.5 + (\frac{2}{3}) * 18 = 47.5 + 12 = 59.5$

3. Final Answer

(i) Grouped Frequency Table: See table in step (i)

(ii) Cumulative Frequency Curves: Plot the curves as described in step (ii)

(iii) Geometric Mean: 59.19

(iv) Median: 65.5

(v) Mode: 59.5