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Gosset’s data on double stout sales: William Sealy Gosset worked at the Guinness Brewery in Dublin and made substantial contributions to the practice of statistics. In his work at the brewery, he collected and analyzed a great deal of data. Archives with Gosset’s handwritten tables, graphs, and notes have been preserved at the Guinness Storehouse in Dublin. In one study, Gosset examined the change in the double stout market before and after World War I (1914–1918). For various regions in England and Scotland, he calculated the ratio of sales in 1925, after the war, as a percent of sales in 1913, before the war. Here are the data:


Bristol 93
Cardiff 113
English Agents 77
English O 68
English P 44
English R 109
Glasgow 65
Liverpool 139
London 427
Manchester 189
Newcastle-on-Tyne 118
Scottish 24

Required:
a. Compute (±0.01) the standard deviation for these data.
b. Compute (±0.01) the quartiles for these data.
c. Which measure do you prefer for describing the spread of this distribution?

Respuesta :

okpalawalter8

Answer:

a

 [tex]\sigma = 101.23[/tex]

b

Lower quartile - [tex]M_1 = 66.5[/tex]

Median  - [tex]M = 101[/tex]

 Upper quartile - [tex]M_2 = 128.5[/tex]

c

Standard deviation

Step-by-step explanation:

From the question we are told that

    The data is  

Bristol 93

Cardiff 113

English Agents 77

English O 68

English P 44

English R 109

Glasgow 65

Liverpool 139

London 427

Manchester 189

Newcastle-on-Tyne 118

Scottish 24

Generally the mean is mathematically represented as

     [tex]\= x = \frac{\sum x_i}{n}[/tex]

Here n is the sample with the value  is  12

   So            

=>         [tex]\= x = \frac{93 + 113 + \cdots + 118 + 24}{12}[/tex]

=>         [tex]\= x = 122.92[/tex]

Generally the standard deviation is mathematically represented as

     [tex]\sigma = \sqrt{\frac{\sum (x_i - \= x)}{n} }[/tex]

=>   [tex]\sigma = \sqrt{\frac{(93 - 122.92)^ 2 + (113 - 122.92) + \cdots + (118 -122.92)^2 + (24 -122.92)^2}{12} }[/tex]

=>     [tex]\sigma = 101.23[/tex]

Generally to compute the quartiles we first arrange the data in ascending order

So

Scottish....................24,  

English P ...................44

Glasgow ..................65

English O .................68

English Agents......... 77

Bristol ..........................93

English R.................... 109

Cardiff......................... 113

Newcastle-on-Tyne.. 118

Liverpool..................... 139

Manchester................. 189

London......................... 427

Next we obtain the median

   Looking at data, the median is  

              [tex]M = \frac{93 + 109}{2}[/tex]

=>            [tex]M = 101[/tex]

Next  is to divide the data in between 93 and  109

So the lower  data set is  

 Scottish....................24,  

English P ...................44

Glasgow ..................65

English O .................68

English Agents......... 77

Bristol ..........................93

The median of the lower data set is  

       [tex]M_1 = \frac{65 + 68}{2}[/tex]

       [tex]M_1 = 66.5[/tex]

This median is known as lower  quartile

Considering the second data set

English R.................... 109

Cardiff......................... 113

Newcastle-on-Tyne.. 118

Liverpool..................... 139

Manchester................. 189

London......................... 427

The median of the upper data set is  

     [tex]M_2 = \frac{118 + 139 }{2}[/tex]

=>  [tex]M_2 = 128.5[/tex]

This median is known as upper  quartile

The best measure for describing the spread of the distribution is the standard deviation  because it measure how far each individual data deviates from the mean