# During the first 13 weeks of the television season, the Saturday eveni

During the first 13 weeks of the television season, the Saturday evening 8:00 P.M. to 9:00 P.M. audience proportions were recorded as ABC 29%, CBS 28% , NBC 24% , and independents 19%. A sample of 300 homes two weeks after a Saturday night schedule revision yielded the following viewing audience data: ABC 95 homes, CBS 64 homes, NBC81 homes, and independents 60 homes. Test with $$\displaystyle{a}={.05}$$ to determine whether the viewing audience proportions changed.
Test statistic =

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Given data,
- Total number of house for sample are 300.
- ABC 95 homes, CBS 64 homes, NBC 81 homes, and independents 60 homes. Test with $$\displaystyle{a}={.05}$$.
- The Saturday evening 8:00 P.M. to 9:00 P.M. audience proportions were recorded as ABC 29%, CBS 28%, NBC 24% , and independents 19%
Step 1
Audience proportions were recorded as ABC 29% , CBS 28% , NBC 24% , and independents 19%.
Taking,
Null Hypothesis $$\displaystyle{H}_{{{0}}}:{P}_{{{A}{B}{C}}}={0.29},{P}_{{{C}{B}{S}}}={0.28},{P}_{{{N}{B}{C}}}={0.24}\ {\quad\text{and}\quad}\ {P}_{{in{d}{e}{p}{e}{n}{d}{e}{n}{t}}}={0.19}$$
Alternative Htpothesis: The population proportion are not,
$$\displaystyle{H}_{{{1}}}:{P}_{{{A}{B}{C}}}={0.29},{P}_{{{C}{B}{S}}}={0.28},{P}_{{{N}{B}{C}}}={0.24}\ {\quad\text{and}\quad}\ {P}_{{in{d}{e}{p}{e}{n}{d}{e}{n}{t}}}={0.19}$$
Here,
Sample size is 100.
Step 2
$$\begin{array}{|c|c|} \hline Category&Proportion&\text{Observated frequency(f)}&\text{Expected frequency(e)}&f-e&(f-e)^{2}&\frac{(f-e)^{2}}{e} \\ \hline ABC&0.29&95&0.29 \times 300=87&8&64&1.73 \\ \hline CBS&0.28&64&0.28 \times 300=84&-20&400&4.76\\ \hline NBC&0.24&81&0.24 \times 300=72&9&81&1.125\\ \hline Independent&0.19&60&0.19 \times 300=57&3&9&0.158\\ \hline &1&300&&&&7.773\\ \hline \end{array}$$
Test statistics:
$$\displaystyle{x}^{{{2}}}=\sum{\frac{{{\left({f}-{e}\right)}^{{{2}}}}}{{{e}}}}$$
$$\displaystyle={7.77}$$