Question

Chapter 2 Exercise is divided in to 2 sections A and B. Data for this assignment is under Data files in module Ex2-30-e8.xls (for A) and Ex2-34-e8.xls (for B). See data under modules.

A) The following data give the weekly amounts spent on groceries for a sample of households.

$271	$363	$159	$ 76	$227	$337	$295	$319	$250
279	205	279	266	199	177	162	232	303
192	181	321	309	246	278	50	41	335
116	100	151	240	474	297	170	188	320
429	294	570	342	279	235	434	123	325

1. Create a frequency table of 9 classes
2. Use the classes as (x) to find the mean of the distribution
3. Use the classes to find the variance and standard deviation -----------------------------------------  (B)The monthly issues of the Journal of Finance are available on the Internet. The table below shows the number of times an issue was downloaded over the last 33 months. Suppose you wish to summarize the number of downloads with a frequency distribution.(This data can be found under data in module titled-Ex2-34-e8.xls

312	2,753	2,595	6,057	7,624	6,624	6,362	6,575	7,760	7,085	7,272
5,967	5,256	6,160	6,238	6,709	7,193	5,631	6,490	6,682	7,829	7,091
6,871	6,230	7,253	5,507	5,676	6,974	6,915	4,999	5,689	6,143	7,086

1. Find the mean, median, mode, minimum and maximum values as well as variance, standard deviation, range, Q1 and Q3.
2. Use the data above to create a frequency distribution table with 11 classes and find the: mean and relative frequency .

Answer 1

Solution

Let x = amount ($) spent on groceries

Back-up Theory

Let x_i =mid-point and f_i be the frequency of ith class_i. Then,

Mean (Average), µ, = Σ(i = 1, n)(x_i.r_i)/Σ(i = 1, n)(f_i) ………………………….......................………. (1)

Variance, σ² = Σ(i = 1, n){f_i.(x_i – µ)²}/ Σ(i = 1, n)(f_i) or equivalently [{Σ(i = 1, n){f_i.(x_i)²} – µ² ]...... . (2)

Standard deviation (SD), σ = sqrt(Variance) ……………………………………........................….. (3)

Now to work out the solution,

Final answers are given below. Back-up Theory and Details of calculations follow at the end.

Part (a)

Frequency Distribution

Maximum value: 570

Minimum value: 41

Range of values: 529

Stipulated number of classes: 9

Class width = 529/9 = 59 ~ 60

Class # i	Class Boundaries	Frequency
1	40 to 100	3
2	100 to 160	4
3	160 to 220	6
4	220 to 280	12
5	280 to 340	10
6	340 to 400	2
7	400 to 460	1
8	460 to 520	1
9	520 to 580	1
	Total	40

Answer 1

Part (b)

Vide (1), mean = 254.5 Answer 2

[Details follow at the end after Part (c).]

Part (c)

Vide (2), variance = 10,509.7500 Answer 3

Vide (3), standard deviation = 102.52 Answer 4

[Details follow]

Class # i	Class Boundaries	Frequency fi	Mid-point xi	fi.xi	di = xi - mean	di2	di2.fi
1	40 to 100	3	70	210	-184.5	34040.25	102120.75
2	100 to 160	4	130	520	-124.5	15500.25	62001
3	160 to 220	6	190	1140	-64.5	4160.25	24961.5
4	220 to 280	12	250	3000	-4.5	20.25	243
5	280 to 340	10	310	3100	55.5	3080.25	30802.5
6	340 to 400	2	370	740	115.5	13340.25	26680.5
7	400 to 460	1	430	430	175.5	30800.25	30800.25
8	460 to 520	1	490	490	235.5	55460.25	55460.25
9	520 to 580	1	550	550	295.5	87320.25	87320.25
	Total	40		10180		243722.25	420390.00

Mean = 10180/40	254.5
Variance = 420390/40	10,509.7500
Standerd deviation = √6093.0563	102.52

DONE