Question

1. 1. A supermarket chain wanted to obtain information on the length of time (in minutes) required to service customers. They need to decide on the number of service counters needed for stores to be built in the future. To find the distribution of customer service times, a sample of 60 customers’ service times were recorded and are shown here:      

3.6        1.9        2.1        .3          .8         .2         1.0        1.4        1.8        1.6

1.4        .2         1.3        3.1        .4         2.3        1.8        4.5        .9         .7

1.6        1.9        5.2        .5          1.8        .3          1.1         .6         .7          .6

Find the mode and the 20^th percentile of the above data set.    

Answer 1

Mode of the given data -

Observation table -

x	frequency
0.2	2
0.3	2
0.4	1
0.5	1
0.6	2
0.7	2
0.8	1
0.9	1
1	1
1.1	1
1.3	1
1.4	2
1.6	2
1.8	3
1.9	2
2.1	1
2.3	1
3.1	1
3.6	1
4.5	1
5.2	1

Mode is the value which is most frequently repeated or which has highest frequency.

Here, we can see in the above table frequency of1.8 is 3, hence -

Mode = 1.8

To find the value of 20th percentile of the above data set, we arrange given data in increasing order -

0.2, 0.2, 0.3, 0.3, 0.4, 0.5, 0.6, 0.6, 0.7, 0.7, 0.8, 0.9, 1.0, 1.1, 1.3, 1.4, 1.6, 1.6, 1.8, 1.8, 1.8, 1.9, 1.9, 2.1, 2.3, 3.1, 3.6, 4.5, 5.2

So,

  

  

  

  

= Size of 6^th observation + 0.2(size of 7^th observation - size of 6^th observation)

= 0.5 + 0.2(0.6 - 0.5)

= 0.5 + 0.02

P₂₀ = 0.52

20^th percentile is 0.52.