Question

Below are the ratings for a sample of 12 hotels in small Ontario towns:

1.1 4.7 4.8 4.8 5.3 6.5 6.7 7.1 7.1 7.3 8.1 9.2

The critical values of this data set are given:

Q1=4.8 Q2=6.6 Q3=7.2

a) Determine the following and show all work: [3 marks]

i) interquartile range (IQR)

ii) upper boundary and lower boundary

iii) any outlier(s)


b) Create a modified box plot for the data, showing any outlier(s). [1 mark]


c) In order for a hotel to be recommended by travel blog TravelWell, its score must be the 80th percentile or higher. What is the minimum rating a hotel can achieve in order to be recommended by TravelWell based on this sample? [2 marks]

Answer 1

a)

i) Interquartile range is 2.4

ii) Upper boundary is 10.8

The Lower boundary is 1.2

iii) Outliers are points above the upper boundary or below the lower boundary

There is one outlier: 1.1

b) The modified boxplot with one outlier is given in the diagram.

c) The 80th percentile is

The minimum rating the hotel can get is 7.3

Explanation:

a)

i) Interquartile range is

IQR = Q3-Q1 = 7.2-4.8 = 2.4

ii) Upper boundary = Q3+ 1.5 *IQR = 7.2+1.5*2.4 = 10.8

Lower boundary = Q1 - 1.5*IQR = 4.8- 1.5*2.4 = 1.2

iii) Outliers are points above the upper boundary or below the lower boundary

There is one outlier: 1.1

c) The 80th percentile is given by

n= 12

k = 0.80

Index is given by k*n =0.80*12 = 9.6

Round to nearest whole number, that is 10th

80th percentile is the 10th number (arranged in ascending order)

80th percentile = 7.3

The minimum rating the hotel can get is 7.3.

b) The modified boxplot with one outlier is given as follows,