Question

A leisure company operates three amusement arcades in the UK: at Redcar, Skegness and Torquay. As part of a performance review the duration in minutes of the period spent in the arcades by each of a sample of customers visiting was recorded.

The durations of visits made by 21 customers visitng the Redcar arcade were :

23 8 39 72 73 13 44 74 37 37 21

21 27 27 34 31 32 43 74 44 36 36 23  

The figures for 18 customers visiting the Skegness arcade were:

31 51 69 12 53 28 36 28 36 30

35 45 48 25 9 32 60 66

The figures for 20 customers visiting the Torquay arcade were:

3 19 1 15 21 9 7 20 10 2

6 2 11 37 10 6 10 14 3 5

(A) Classify both sets of data into grouped frequency distributions

(B) Calculate the relative frequency for each class of all three distributions

(C) The company expects customers to spend at least 20 minutes on visits to their arcades.

Use your relative frequency figures to compare the performances of the arcades in this respect.

Answer 1

(a) Redcar Arcade:

The maximum of the data=74

Min of data=8

Range=Max-Min=74-8=66

Let the number of class intervals be 6

Then the width of each class=66/6=11

We construct the following frequency table:

Class Interval	Frequency
8-19	2
20-31	7
32-43	6
44-55	2
56-67	0
68-79	4
Total	21

Skegness Arcade:

The maximum of the data=69

Min of data=9

Range=Max-Min=69-9=60

Let the number of class intervals be 6

Then the width of each class=60/6=10

We construct the following frequency table:

Class Interval	Frequency
9-19	2
20-30	4
31-41	5
42-52	3
53-63	2
64-74	2
Total	18

Torquay Arcade:

The maximum of the data=37

Min of data=1

Range=Max-Min=37-1=36

Let the number of class intervals be 6

Then the width of each class=36/6=6

We construct the following frequency table:

Class Interval	Frequency
1-7	9
8-14	6
15-21	4
22-28	0
29-35	0
36-42	1
Total	20

(b) We calculate Relative Frequency for the above three:

Relative Freq = Freq/Total

Redcar Arcade:

Table 1:

Class Interval	Frequency	Relative Frequency
8-19	2	0.0952
20-31	7	0.3333
32-43	6	0.2857
44-55	2	0.0952
56-67	0	0.0000
68-79	4	0.1905
Total	21	1

Skegness Arcade:

Table 2:

Class Interval	Frequency	Relative Frequency
9-19	2	0.111
20-30	4	0.222
31-41	5	0.278
42-52	3	0.167
53-63	2	0.111
64-74	2	0.111
Total	18	1

Torquay Arcade:

Table 3:

Class Interval	Frequency	Relative Frequency
1-7	9	0.45
8-14	6	0.3
15-21	4	0.2
22-28	0	0
29-35	0	0
36-42	1	0.05
Total	20	1

(c) The customers spend atleast 20 minutes is the total relative frequency for the class greater than or equal to 20.

So for Redcar Arcade:

From Table 1:

Total of relative frequencies for class greater than equal to 20 =0.3333+0.2857+0.0952+0+0.1905=0.9048

From Table 2:

Total of relative frequencies for class greater than equal to 20 =0.222+0.278+0.167+0.111+0.111=0.889

From Table 3:

Since 20 is in the class 15-21 we need to calculate relative frequency for greater than equal to 20.

We see the Total of relative frequencies for class greater than equal to 22=0+0+0.05=0.05

From the data,we see the number of observations between 20-21 is , the relative frequency is therefore =2/20=0.1

Hence Total of relative frequencies for class greater than equal to 20 =0.05+0.1=0.15

Hence, the highest is observed from Table 1 i.e, Redcar Arcade is the highest among all three.

So for customers to spend at least 20 minutes ,

Redcar Arcade > Skegness Arcade > Torquay Arcade