Question

The number of annual precipitation days for one-half of the 50 largest U.S. cities is listed below. Do the following:

131 94 136 88 116 77 127 79 47 97 116 123 88 102 27 80 156 133 117 55 112 98 55 90 125

a) Construct a frequency distribution for the data (include class boundaries in the table) with 5 classes.

b) Graph its histogram.

c) Plot its cumulative frequency (ogive).

d) Find the mode of the data.

e) Find the mean, , for the grouped data.

f) Find the variance and standard deviation for grouped data.

g) Find the z score and percentile for the value 98.

Answer 1

Solection: ascending order Ket us frost arrange the values in 27,47,55,55,77, 79, 80, 88, 89, 90,94, 97, 98,102,112,116,116, 117, 123, 125, 127, 133, 133, 136, 156 (a) Since we are given total of 25 values and to create 5 elasses it is obvious to create classes with width 12XX16 elass interval Class width frequency (fi) 25 - 54 class boundaries 24.5-54-5 54-5-84.5 30 30 55 - 84 5 30 85 - 114 8 84.5 - 114-5 30 115 - 144 9 114.5.-144.5 30 (5 145 - 174 1 144-5-174-5 N=2fe= 25 SL- Cumulative freq.Cless than) Cumulative free. (more than) 2 25 -7 23 15 18 24 10 25 1 SL frequency density fjelass with Class:mark (21) 39.5 007 0.17 69.5 0-27 99.5 129.5 0.30 159.5 003 Toe (6) To draw the histogram plot frequency de densities in y-axis and elass boundaries in x-axis. (diagram obtained through ezeel)

plot, less than type yaxis (for more than type ogive coe frequencies in y axis) (diagram obtained through czeel). (d) mode is the that value of the data sel which (e) To draw less than type ogive, cumulative frequencies in plot more than type eumcelotive and elass boundaries in x axis these too ogives interseet at median oceurs maximcem times. here 55,88 and 116 are mode. (e) mean, Kiti (39-5* 2) + (69-5*5) +(99-5x8) +(129-539) +(159.5x1) 25 101.9. 1 (f) Varianee, 52 xfi - 52 Nil 930. 24. and standard deviation, s = 82 1930.24 = 30-4998. upp ZxBxgøre -9) for value 98, Z-score. value - mean standard deviation (98 - 101.9) 30.4998 -0-1278: Suppose; =A8. Voo 1727 98 is the eth percentile value (say). Pe Cx25 - 7 98 = 84.5+ 100 X 30 = e = 42.4 8

HISTOGRAM 0.3 0.2 frequency density 0.1 0.0 24.5-54.5 54.5-84.5 84.5-114.5 114.5-144.5 144.5-174.5 class boundary
Though, in histogram and ogive, the class boundaries in x axis should be laterally side by side, we zoomed in and created gap intentionally for better understanding purpose of the figure.

LESS THAN TYPE OGIVE 25 20 15 cumulative frquency (less than type) 10 24.5-54.5 54.5-84.5 84.5-114.5 114.5-144.5 144.5-174.5 class boundary

GREATER THAN TYPE OGIVE 25 20 15 cumulative frequency (greater than type) 10 24.5-54.5 54.5-84.5 84.5-114.5 114.5-144.5 144.5-174.5 class boundary