Question

The Los Angeles Times regularly reports the air quality index for various areas of Southern

California. A sample of air quality index values for Pomona provided the following
data: 28, 42, 58, 48, 45, 55, 60, 49, and 50.
a. Compute the range and interquartile range.
b. Compute the sample variance and sample standard deviation.
c. A sample of air quality index readings for Anaheim provided a sample mean of 48.5,a sample variance of 136, and a sample standard deviation of 11.66. What comparisons can you make between the air quality in Pomona and that in Anaheim on the basis of these descriptive statistics?

Answer 1

Range = 32 ; IQR = 13 ; Variance = 92.75 ; Standard deviation = 9.63

Explanation:

Given the data:

28, 42, 58, 48, 45, 55, 60, 49, 50

Reorderd data: 28, 42, 45, 48, 49, 50, 55, 58, 60

The range: maximum - minimum

Range = 60 - 28 = 32

Interquartile range (IQR) : Q3 - Q1

Q3 = 3/4(n+1) th term. ; n = sample size

Q1 = 1/4(n+1)th term.

Using calculator :

Q3 = 56.5 ; Q1 = 43.5

IQR = 56.5 - 43.5 = 13

The variance :

Σ(X - m)^2 / n-1

m = Σx / n

Mean, m = 435 / 9 = 48.33

Variance :

[(28-48.33)^2 + (42-48.33)^2 + (45-48.33)^2 + (48-48.33)^2 + (49-48.33)^2 + (50-48.33)^2 + (55-48.33)^2 + (58-48.33)^2 + (60-48.33)^2] / 8

= 742.0001 / 8

= 92.75

Standard deviation = sqrt(Variance):

Sqrt(92.75) = 9.63

C.)

Sample mean are very close with a slightly greater Variability in the air quality of Anaheim over Pomona