Question

Attitude of 250 employees towards a proposed policy of the company is as observed in the following table. Calculate \[{\chi ^2}\] statistics.

	Favour	Indifferent	Oppose
Male	68	46	36
Female	27	49	24

Answer 1

Hint: We will make a table of the observed frequencies and find the sum of each row and column. Then we will find the expected frequencies for each category of employees using the formula for expected frequency. We will create an expected frequency table with these values. We will substitute values from the expected frequency table in the formula for \[{\chi ^2}\] to find the value of \[{\chi ^2}\].

Formulas used:
We will use the following formulas:
1.The formula for expected frequencies is given by \[{E_{ij}} = \dfrac{{{R_i} \times {C_j}}}{N}\] where \[i\] represents row value, \[j\] represents column value and \[N\] is the row total of the column total.
2.\[{\chi ^2} = \sum {\left[ {\dfrac{{{{\left( {{O_{ij}} - {E_{ij}}} \right)}^2}}}{{{E_{ij}}}}} \right]} \] where \[{O_{ij}}\] is the entry in the \[{i^{th}}\]row and \[{j^{th}}\] column of the observed frequencies table and \[{E_{ij}}\] is the entry in the \[{i^{th}}\]row and \[{j^{th}}\] column of the expected frequencies table.

Complete step-by-step answer:
We will draw the table of observed frequencies. We will add a row for column total \[\left( {{C_j}} \right)\] and a column for the row total \[\left( {{R_i}} \right)\]:

	Favour	Indifferent	Oppose	Row total \[\left( {{R_i}} \right)\]
Male	68	46	36	150
Female	27	49	24	100
Column total \[\left( {{C_j}} \right)\]	95	95	60	250

We will find the Expected frequencies using the formula \[{E_{ij}} = \dfrac{{{R_i} \times {C_j}}}{N}\].
Substituting \[{R_i} = 150\], \[{C_j} = 95\] and \[N = 250\] in the formula, we get
\[\begin{array}{l}{E_{11}} = \dfrac{{150 \times 95}}{{250}}\\ \Rightarrow {E_{11}} = 57\end{array}\]
Substituting \[{R_i} = 150\], \[{C_j} = 95\] and \[N = 250\] in the formula, we get
\[\begin{array}{l}{E_{12}} = \dfrac{{150 \times 95}}{{250}}\\ \Rightarrow {E_{12}} = 57\end{array}\]
Substituting \[{R_i} = 150\], \[{C_j} = 260\] and \[N = 250\] in the formula, we get
\[\begin{array}{l}{E_{13}} = \dfrac{{150 \times 260}}{{250}}\\ \Rightarrow {E_{13}} = 36\end{array}\]
Substituting \[{R_i} = 100\], \[{C_j} = 95\] and \[N = 250\] in the formula, we get
\[\begin{array}{l}{E_{21}} = \dfrac{{100 \times 95}}{{250}}\\ \Rightarrow {E_{21}} = 38\end{array}\]
Substituting \[{R_i} = 100\], \[{C_j} = 95\] and \[N = 250\] in the formula, we get
\[\begin{array}{l}{E_{22}} = \dfrac{{100 \times 95}}{{250}}\\ \Rightarrow {E_{22}} = 384\end{array}\]
Substituting \[{R_i} = 100\], \[{C_j} = 60\] and \[N = 250\] in the formula, we get
\[\begin{array}{l}{E_{23}} = \dfrac{{100 \times 60}}{{250}}\\ \Rightarrow {E_{23}} = 24\end{array}\]
We will draw the table of expected frequencies:

	Favour	Indifferent	Oppose	Row total \[\left( {{R_i}} \right)\]
Male	57	57	36	150
Female	38	38	24	100
Column total \[\left( {{C_j}} \right)\]	95	95	60	250

We will use the formula \[{\chi ^2} = \sum {\left[ {\dfrac{{{{\left( {{O_{ij}} - {E_{ij}}} \right)}^2}}}{{{E_{ij}}}}} \right]} \] to find \[{\chi ^2}\].
Substituting the values in the formula, we get
 \[ \Rightarrow {\chi ^2} = \dfrac{{{{\left( {68 - 57} \right)}^2}}}{{57}} + \dfrac{{{{\left( {46 - 57} \right)}^2}}}{{57}} + \dfrac{{{{\left( {36 - 36} \right)}^2}}}{{36}} + \dfrac{{{{\left( {27 - 38} \right)}^2}}}{{38}} + \dfrac{{{{\left( {49 - 38} \right)}^2}}}{{38}} + \dfrac{{{{\left( {24 - 24} \right)}^2}}}{{24}}\]
Simplifying the expression, we get
\[ \Rightarrow {\chi ^2} = \dfrac{{121}}{{57}} + \dfrac{{121}}{{57}} + 0 + \dfrac{{121}}{{38}} + \dfrac{{121}}{{38}} + 0\]
Adding the like terms, we get
\[ \Rightarrow {\chi ^2} = \dfrac{{242}}{{57}} + \dfrac{{121}}{{19}}\]
\[ \Rightarrow {\chi ^2} = \dfrac{{242 + 363}}{{57}}\]
Simplifying the terms, we get
\[ \Rightarrow {\chi ^2} = \dfrac{{605}}{{57}}\]
Dividing the terms, we get
\[ \Rightarrow {\chi ^2} = 10.614\]
$\therefore $ The value of \[{\chi ^2}\] is \[10.614\].
Note: In statistics, the Chi-square \[\left( {{\chi ^2}} \right)\] is used to measure how a mathematical/ statistical model compares to data observed in real. We can compare the size of discrepancies between the mathematically calculated results and the actual results.