Question

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5. The market shares in 2012 of four (4) television companies, \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\) are listed below. A survey was camied out in 2017 on 200 television buyers and the distribution of buyers to \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\) are also listed in the last column.

$$ \begin{array}{|c|c|c|} \hline \text { Company } & \text { Market share in 2012 } & \text { Number of buyer in 2017 } \\ \hline \text { A } & 0.40 & 70 \\ \hline \text { B } & 0.32 & 60 \\ \hline \text { C } & 0.24 & 54 \\ \hline \text { D } & 0.04 & 16 \\ \hline \end{array} $$

(a) Specify the competing hypotheses to test whether the market shares have changed since 2012.

(b) Calculate the value of the test statistics.

(c) Use \(\alpha=0.05\) to determine if the market shares have changed since 2012 .

Answer 1

To test the hypothesis, the market shares changed in 2012 at a 5% level of significance.

(a)

The null and alternative hypothesis is :

The null hypothesis is that the market shares did not change in 2012.

The alternative hypothesis is that the market shares changed in 2012.

(b)

The chi-square test statistics,

First, compute the frequency expected, then find chi-square test statistics.

The expected frequency for Company A is,

$$ \begin{aligned} f_{e} &=N p i \\ &=200 \times 0.40 \\ &=80 \end{aligned} $$

The expected frequency for Company \(\mathrm{B}\) is,

$$ \begin{aligned} f_{e} &=N p i \\ &=200 \times 0.32 \\ &=64 \end{aligned} $$

The expected frequency for Company \(\mathrm{C}\) is,

$$ f_{e}=N p i $$

$$ \begin{array}{l} =200 \times 0.24 \\ =48 \end{array} $$

The expected frequency for Company D is,

$$ \begin{aligned} f_{e} &=N p i \\ &=200 \times 0.04 \\ &=8 \end{aligned} $$

The chi-square test statisticis,

$$ \chi^{2}=\sum \frac{\left(f_{o}-f_{e}\right)^{2}}{f_{e}} $$

$$ \begin{array}{l} =\frac{(70-80)^{2}}{80}+\frac{(60-64)^{2}}{64}+\frac{(54-48)^{2}}{48}+\frac{(16-8)^{2}}{8} \\ =10.250 \end{array} $$

The chi - square test statistic is 10.250

(c)

The chi - square critical value is,

First, compute degrees of freedom for finding the chi-square critical value.

The degrees of freedom are,

$$ \begin{aligned} \text { d. } f . &=k-1 \\ &=4-1 \\ &=3 \end{aligned} $$

The chi-square critical value table shows degrees of freedom 2 and the significance level 0.05, so the chi-square critical value is 7.815.

The chi-square critical value is 7.815

Decision

The conclusion is that the chi-square test statistic 10.250 is greater than the chi-square critical value 7.815, so the null hypothesis is rejected at a 5% level of significance. There is sufficient evidence to indicate that the market shares changed in 2012. The result is statistically significant.