Question

A sample of boys and girls were asked to choose one color from the three options – pink, blue, and orange to paint their room with the following results. Calculate \[{{\chi }^{2}}\] statistics.

	Pink	Blue	Orange
Boys	27	63	10
Girls	41	45	14

Answer 1

Hint: We solve this problem by using the direct formula for \[{{\chi }^{2}}\] which is given as
\[{{\chi }^{2}}=\sum{\dfrac{{{\left( O-E \right)}^{2}}}{E}}\]
Where, \['O'\] is the observed frequency and \['E'\] is the frequency of expected value of corresponding observed value which is calculated by using the row total times the column total times divided by grand total. That is for a given data as follows

	Category I	Category II	Category III	Rows total
Sample A	a	b	c	a+ b+ c
Sample B	d	e	f	d+ e+ f
Column total	a+ d	b+ e	c+ f	N=a+ b+ c+ d+ e+ f

Here, for the observed value ‘a’ the expected value is given as
\[{{E}_{11}}=\dfrac{\left( a+b+c \right)\left( a+d \right)}{N}\]
By using this formula we calculate the \[{{\chi }^{2}}\] for the given data.

Complete step-by-step solution:
Let us assume the given data and add one row and column by adding the rows and columns then we get

	Pink	Blue	Orange	Rows total
Boys	27	63	10	100
Girls	41	45	14	100
Column total	68	108	24	200 = N

Now let us calculate the expected values for all the observed values.
We know that the expected value is calculated by using the row total times the column total times divided by grand total. That is for a given data as follows

	Category I	Category II	Category III	Rows total
Sample A	a	b	c	a+ b+ c
Sample B	d	e	f	d+ e+ f
Column total	a+ d	b+ e	c+ f	N=a+ b+ c+ d+ e+ f

Here, for the observed value ‘a’ the expected value is given as
\[{{E}_{11}}=\dfrac{\left( a+b+c \right)\left( a+d \right)}{N}\]
Let us take the first observed value that is ‘27’ then we get the expected value as
\[\Rightarrow {{E}_{11}}=\dfrac{100\times 68}{200}=34\]
Now let us take the second observed value from first row then we get the expected value as
\[\Rightarrow {{E}_{12}}=\dfrac{100\times 108}{200}=54\]
Similarly, for the third observed value in first row we get the expected value as
\[\Rightarrow {{E}_{13}}=\dfrac{100\times 24}{200}=12\]
Now, similarly let us calculate the expected values for the observed values in second row then we get
\[\Rightarrow {{E}_{21}}=\dfrac{100\times 68}{200}=34\]
\[\Rightarrow {{E}_{22}}=\dfrac{100\times 108}{200}=54\]
\[\Rightarrow {{E}_{23}}=\dfrac{100\times 24}{200}=12\]
Now, we can see that we have got all the expected values for the corresponding observed values.
We know that the formula for \[{{\chi }^{2}}\] which is given as
\[{{\chi }^{2}}=\sum{\dfrac{{{\left( O-E \right)}^{2}}}{E}}\]
By substituting all the required observed and corresponding expected values we get
\[\begin{align}
  & \Rightarrow {{\chi }^{2}}=\dfrac{{{\left( 27-34 \right)}^{2}}}{34}+\dfrac{{{\left( 63-54 \right)}^{2}}}{54}+\dfrac{{{\left( 10-12 \right)}^{2}}}{12}+\dfrac{{{\left( 41-34 \right)}^{2}}}{34}+\dfrac{{{\left( 45-54 \right)}^{2}}}{54}+\dfrac{{{\left( 14-12 \right)}^{2}}}{12} \\
 & \Rightarrow {{\chi }^{2}}=\dfrac{49}{34}+\dfrac{81}{54}+\dfrac{4}{12}+\dfrac{49}{34}+\dfrac{81}{54}+\dfrac{4}{12} \\
\end{align}\]
Now, by regrouping the terms with same denominator we get
\[\begin{align}
  & \Rightarrow {{\chi }^{2}}=\left( \dfrac{49}{34}+\dfrac{49}{34} \right)+\left( \dfrac{81}{54}+\dfrac{81}{54} \right)+\left( \dfrac{4}{12}+\dfrac{4}{12} \right) \\
 & \Rightarrow {{\chi }^{2}}=\dfrac{49}{17}+3+\dfrac{2}{3} \\
\end{align}\]
By taking the LCM and adding the fractions then we get
\[\begin{align}
  & \Rightarrow {{\chi }^{2}}=\dfrac{147+153+34}{51} \\
 & \Rightarrow {{\chi }^{2}}=\dfrac{334}{51}=6.56 \\
\end{align}\]
Therefore the value of chi square \[{{\chi }^{2}}\] for given data is 6.56.

Note: Here, for the easy calculations we can create a table such that we can reduce the calculations for the time being.
Let us create a table containing the observed values and corresponding expected values that are shown below.

Observed value (O)	Expected values (E)	\[\left| O-E \right|\]	\[{{\left( O-E \right)}^{2}}\]	\[\dfrac{{{\left( O-E \right)}^{2}}}{E}\]
27	34	7	49	1.44
63	54	9	81	1.5
10	12	2	4	0.34
41	34	7	49	1.44
45	54	9	81	1.5
14	12	2	4	0.34

Now the value of chi square \[{{\chi }^{2}}\] is given by adding all the values on last column that is
\[\begin{align}
  & \Rightarrow {{\chi }^{2}}=1.44+1.5+0.34+1.44+1.5+0.34 \\
 & \Rightarrow {{\chi }^{2}}=6.56 \\
\end{align}\]
Therefore the value of chi square \[{{\chi }^{2}}\] for given data is 6.56.