Question

During the covid-19 pandemic, a fast food company in Macau recorded their sales in one day (in proper units) of 40 days as the following.  

72	70	47	54	51	64	68	68	59	77
67	59	60	59	71	46	69	58	76	53
60	74	57	25	53	52	67	65	47	49
72	60	53	78	71	63	51	80	70	59

a. Find the mode and median of the data.

b. Determine the first and third quartiles, and find the interquartile range.

c. Find the five-number summary.

d. Lower limit and upper limit of the data.

e. Find the outlier(s) with reasons, if any.

Process,pls

Answer 1

Arrange the given data in sorted order and assign positional values as shown below:

Values in sorted order	Position or rank
25	1
46	2
47	3
47	4
49	5
51	6
51	7
52	8
53	9
53	10
53	11
54	12
57	13
58	14
59	15
59	16
59	17
59	18
60	19
60	20
60	21
63	22
64	23
65	24
67	25
67	26
68	27
68	28
69	29
70	30
70	31
71	32
71	33
72	34
72	35
74	36
76	37
77	38
78	39
80	40

The first half of the data set represent values corresponding to position 1 to 20.

The second half of the data set represent values corresponding to 21st to 40th position.

(a).

Mode: The most frequently occurred value in the data set is 59. Therefore, the mode is 59.

Median: The median value equals to the average of values corresponding to 20th and 21st position in the sorted data set.

The values at 20th and 21st position are 60 and 60 respectively.

Therefore, the median is 60.

(b).
The first quartile (Q1) is the average of values corresponding to 10th and 11th position in the first half of the sorted data set.

The values at 10th and 11th position are 53 and 53 respectively.

Therefore, the first quartile (Q1) is 53.

The third quartile (Q3) is the average of values corresponding to 30th and 31st position in the second half of the sorted data set.​​​​​​​

The values at 30th and 31st position are 70 and 70 respectively.

Therefore, the third quartile (Q3) is 70

The interquartile range (IQR) is

IQR = Q3 – Q1 = 70 – 53 = 17

Therefore, IQR is 17.

(c).

The five-number summary is,

Minimum	25
First Quartile	53
Second Quartile or Median	60
Third Quartile	70
Maximum	80

(d). The lower and upper limit of the data can be obtained by using 1.5 IQR rule can be calculated as,

Therefore, the lower limit is 27.5 and upper limit is 95.5.

(e).

1.5 IQR rule can be used to determine the presence of outliers in the data set.

From the part (d), it can be seen that there is only one value; that is, 25 which does not fall within the limits (27.5, 95.5), so 25 is an outlier in the data set.

Therefore, the outlier in the data set is 25.