Question

(the solution for this question is not available, please answer complete)

Second, the researcher wishes to use graphical descriptive methods to present summaries of the data on each of the two variables: hours worked per week and yearly income, as stored in file HOURSWORKED.xls. a) The number of observations (n) is 65 individuals. The researcher suggests using 7 class intervals to construct a histogram for each variable. Explain how the researcher would have decided on the number of class intervals (K) as 7. b) The researcher suggests using class intervals as 10 < X ≤ 15, 15 < X ≤ 20, …, 40 < X ≤ 45 for the hours per week variable and class intervals 40 < X ≤ 45, 45 < X ≤ 50, ..., 70 < X ≤ 75 for the yearly income variable. Explain how the researcher would have decided the width of the above class intervals (or class width). c) Draw and display a histogram for each of the two variables using appropriate BIN values from part (b) and comment on the shape of the two distributions.

Hours Per Week	Yearly Income ('000's)
18	43.8
13	44.5
18	44.8
25.5	46.0
11.5	41.2
18	43.3
16	43.6
27	46.2
27.5	46.8
30.5	48.2
24.5	49.3
32.5	53.8
25	53.9
23.5	54.2
30.5	50.5
27.5	51.2
28	51.5
26	52.6
25.5	52.8
26.5	52.9
33	49.5
15	49.8
27.5	50.3
36	54.3
27	55.1
34.5	55.3
39	61.7
37	62.3
31.5	63.4
37	63.7
24.5	55.5
28	55.6
19	55.7
38.5	58.2
37.5	58.3
18.5	58.4
32	59.2
35	59.3
36	59.4
39	60.5
24.5	56.7
26	57.8
38	63.8
44.5	64.2
34.5	55.8
34.5	56.2
40	64.3
41.5	64.5
34.5	64.7
42.3	66.1
34.5	72.3
28	73.2
38	74.2
31.5	68.5
36	69.7
37.5	71.2
22	66.3
33.5	66.5
37	66.7
43.5	74.8
20	62.0
35	57.3
24	55.3
20	56.1
41	61.5

Answer 1

a) -

The number of observations is 65 indivisuals. So, n = 65. We can calculate number of classes by the formula -   

Number of classes = 1 + 3.322log10n = 1 + 3.322log10(65) = 1 + 3.3221.8129 = 7.0224 7

Hence, number of classes can be taken as 7.

b) -

Class interval for the can be calculated by the formula -

Where, i = class interval, L = largest observation , S= smallest observation , n = total number of observations

Class interval for the hours per week -

L = largest observation = 44.5 , S= smallest observation = 11.5 , n = total number of observations = 65

So, the class interval can be taken as 5. Hence, the classes becomes as 10<x15,15<x20,.......,40<x45.

Class interval for the yearly income -

L = largest observation = 74.2 , S= smallest observation = 41.8 , n = total number of observation

So, the class interval can be taken as 5. Hence, the classes becomes as 40<x45,45<x50,.......,70<x75.

c)

Frequency distribution for the hours per week -

Class	frequency
10 -15	3
15-20	8
20-25	7
25-30	13
30-35	15
35-40	14
40-45	5
Total	65

histogram for hours per week -

Comment - Shape of the distribution seems to be skewed to the left, so the shape of the distribution is negatively skewed.

Frequency distribution for yearly income -

Class	frequency
40-45	6
45-50	7
50-55	11
55-60	18
60-65	12
65-70	6
70-75	5
Total	65

Histogram for yearl income -

Comment - Shape of the distribution seems to be bell shaped, so shape of distribution is symmetric.