Question

The Los Angeles Times regularity reports the air quality index for various areas os Southern California. A sample of air quality index values for San Bernardino provides the following data: 56, 84, 116, 96, 90, 110, 120, 98, and 100

a) compute the mean, median, and the mode of the data observed and interpret

b) compute the first quartile and interpret

c) compute the range

d) compute the sample variance and sample standard deviation

e) A sample of air quality index reading for Anaheim provided a sample mean of 97, and a sample standard deviation of 18.23. If someone health is affected by the air quality and its variability where it would be better for him or her to live

Answer 1

In this problem we are given the data on the air quality index for the various areas of Southern California.

a> To compute the mean we need to divide the sum of the values by the total number of values.

Mean= (56+84+116+96+90+110+120+98+100)/9 = 96.667 ( correct upto 3 decimal places)

Median is the middlemost value in a given set of observations. To compute the median we first need to arrange the sample values in their ascending order and pick up the middlemost value.

The observations in increasing order are : 56,84,90,96,98,100,110,116,120.

Median= 98

The mode is the highest value in a given set of values where all of them are distinct.

Mode= 120

Interpretation: The mean is the most appropriate among the three measures of Central Tendency. It indicates that the air quality index of Southern California is 96.667 on an average which throws some light on the given data.

b> Let x1, x2, ......, x9 be the 9 values arranged in ascending order of magnitude.

x1<x2<......<x9

The observations in increasing order are : 56,84,90,96,98,100,110,116,120.

The first quartile is nothing but  

Here, p= 1/4 and n =9

n*p=9/4 and n(1-p)= 9(1-(1/4)) =27/4

Note that, { No of Xi which are less than equal to x3} =3 > 9/4

{ No of Xi which are greater than equal to x3} =7 > 9 (1-1/4) =27/4

Hence , x3 is the first quartile i.e. = 90

Interpretation :  About 25 % of the values in the given data lie below 90 and the remaining 75% above 90.

c> The range of a set of values is the difference between the maximum and the minimum values.

In the given problem, max { Xi} =120 and min{Xi} =56

So, Range = 120-56 =64

d> To compute the sample variance , we first need to compute the differences (xi - ) for all i , i= 1(1)9

The differences are calculated as : -40.667, -12.667, 19.333, -0.667, -6.667, 13.333, 23.333, 1.333 ,3.333

Now we compute , =2968

Hence, Variance = 2968/9 =329.778 ( correct upto 3 decimal places)

Now,  Standard deviation is the positive square root of variance

Standard deviation=18.16 ( correct upto 3 decimal places)