Question

The Staten Island Times regularly reports the air quality index for various areas of New York. A sample of air quality index values for Harlem provided the following data: 32, 28, 26, 35, 43, 62, 57, 49, and 67.

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 Queens provided a sample mean of 43.5, a sample variance of 126, and a sample standard deviation of 9.46. What comparisons can you make between the air quality in Harlem and that in Queens on the basis of these descriptive statistics?

Answer 1

a) The data in the ascending order is  26 28 32 35 43 49 57 62 67.

So, range = 67 - 26 = 41

Q1 = 32 , because the number of observation less than or equal to 32 is 3. (25% of 9 = 2.25)

Q3 = 57 , because the number of observation less than or equal to 57 is 7. (75% of 9 = 6.75)

So, interquartile range = 57 - 32 = 25

b) Using the formula for we get sample mean as 44.33 and sample variance as 231.5

so, sample sd = sqrt(231.5) = 15.215

Sample mean

c) Kindly, note that there is a problem in the question, if sample variance of Queens is 126, then the sample standard deviation will be square root of 126 = 11.22, it can't be 9.46, there is a printing mistake in either if variance or standard deviation.  

Assuming sample SD is given correctly, we see that mean air quality index for Queens is 43.5 which is less than that for Harle, and also, the sample standard deviation for Harlem is greater than Queen, so the AQI for Queens is not only better on an average, also, it is more consistent than Harlem.