Question

survey is carried out to study the number of hours, X, per day spent on using the Internet by the customers of an Internet Service Provider (ISP). Responses from 15 randomly selected customers give the following data:       3    2    5    6    1    5    3    4    6    2    4    5    6    5    1

(a) Find out the five‐number summary and sketch a boxplot. Describe the distribution shape.

(b) Suppose a selected customer is found to spend at most 5 hours on using the Internet. What is the probability that this customer spends more than 3 hours on using the Internet?  

(c) Suppose the customer service manager of the ISP discovers that one data point has been mistakenly recorded to 4, and that data pointshould be lessthan 4. State whether the mode, median and mean of the correct sample become smaller than, greater than or the same as those of the incorrect sample. Briefly explain.

Answer 1

Ans) a) The R code for five number summary and the boxplot is given by:

x=c(3,2,5,6,1,5,3,4,6,2,4,5,6,5,1)
summary(x)
boxplot(x)

The five number summary is coming to be :

Minimum value : 1 , 1st Quartile : 2.50 , Median : 4 , 3rd quartile : 5 , Maximum value : 6

The boxplot is given by:

From the above graph we can observe that the median 4 is slightly upwards. So the lower portion of the box is larger. So we can conclude that the shape of the distribution is skewed left or positively skewed.

b) Given a customer spends at most 5 hours on internet. We have to find the probability that he will spend more than 3 hours on internet. So we have to find P(3<X<=5) . We can calculate this using distribution of all the customer spending hours on internet. There are two customers spend 4 hours on internet and 4 customers spend 5 hours on internet. So there are total 6 customers who spends more than 3 hours and at most 5 hours on internet. There are total 15 customers. So the desired probability will be 6/15 = 2/5.

c) Given one customer who is given spending 4 hours actually spends less than that. The mean is given by $\frac{1}{n}\sum_{i=1}^{n}X_i$ So it depends on each of the values of the data. If one data value 4 changes to less than 4 then mean is going to change and it will decrease as the value 4 is decreased.

The median is the middle most value of the data. It indicate the value of which 50% of the data is less than that and 50% is higher than that. If one data that indicates 4, is less than that then the median is still remain 4 because there is another 4. So the value of median will remain same because at this case 4 still represents the middle most value or 50% of the data is less than or greater than that value 4.

The mode will also remain same. Because mode is the most likely value of the data. Here it is 5. It occurs 4 times. If one of the 4 becomes becomes 1,2 or 3 then the frequency of them becomes 2 to 3 which is still less than 4. So here in this case also the most likely value is 5. So the mode remains 5.