The sequence is as follows:
A | N | A | N | N | N | A | A | A | A | N | A | A | A | A | N | A | N | N | A | A | N | N | A |
A | A | A | N | A | N | N | A | A | A | A | A | N | A | N | A | N | A | N | A | A | A | A | A |
A | A | N | N | A | N | A | N | N | A | A | N | N | N | A | N | A | N | A | N | A | A | A | N |
N | A | A | N | N | N | N | A | A | A | N | A | N | A | N | A | A | A | N | A | N | A | A | A |
N | A | N | A | A | N | A | N | A | N | N | N | A | N |
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Now, the number of times N occurs is denoted by\({n_1}\)and the number of times A occurs is denoted by\({n_2}\).
Thus,
\(\begin{array}{l}{n_1} = 47\\{n_2} = 63\end{array}\)
The runs of the sequence are formed as follows:
\(\underbrace A_{{1^{st}}run}\underbrace N_{{2^{nd}}run}\underbrace A_{{3^{rd}}run}\underbrace {NNN}_{{4^{th}}run}\underbrace {AAAA}_{{5^{th}}run}\underbrace N_{{6^{th}}run}\underbrace {AAAA}_{{7^{th}}run}\underbrace N_{{8^{th}}run}\underbrace A_{{9^{th}}run}\underbrace N_{{{10}^{th}}run}\underbrace {AA}_{{{11}^{th}}run}\underbrace {NN}_{{{12}^{th}}run}\underbrace A_{{{13}^{th}}run}\)
\(\underbrace {AA}_{{{14}^{th}}run}\underbrace N_{{{15}^{th}}run}\underbrace A_{{{16}^{th}}run}\underbrace {NN}_{{{17}^{th}}run}\underbrace {AAAAA}_{{{18}^{th}}run}\underbrace N_{{{19}^{th}}run}\underbrace A_{{{20}^{th}}run}\underbrace N_{{{21}^{st}}run}\underbrace A_{{{22}^{nd}}run}\underbrace N_{{{23}^{rd}}run}\underbrace A_{{{24}^{th}}run}\underbrace N_{{{25}^{th}}run}\underbrace {AAAAA}_{{{26}^{th}}run}\)
\(\underbrace A_{{{27}^{th}}run}\underbrace {NN}_{{{28}^{th}}run}\underbrace A_{{{29}^{th}}run}\underbrace N_{{{30}^{th}}run}\underbrace A_{{{31}^{st}}run}\underbrace {NN}_{{{32}^{nd}}run}\underbrace {AA}_{{{33}^{rd}}run}\underbrace {NNN}_{{{34}^{th}}run}\underbrace A_{{{35}^{th}}run}\underbrace N_{{{36}^{th}}run}\underbrace A_{{{37}^{th}}run}\underbrace N_{{{38}^{th}}run}\underbrace A_{{{39}^{th}}run}\underbrace N_{{{40}^{th}}run}\underbrace {AA}_{{{41}^{st}}run}\underbrace N_{{{42}^{nd}}run}\)
\(\underbrace {AA}_{{{43}^{rd}}run}\underbrace {NNNN}_{{{44}^{th}}run}\underbrace {AAA}_{{{45}^{th}}run}\underbrace N_{{{46}^{th}}run}\underbrace A_{{{47}^{th}}run}\underbrace N_{{{48}^{th}}run}\underbrace A_{{{49}^{th}}run}\underbrace N_{{{50}^{th}}run}\underbrace {AAA}_{{{51}^{th}}run}\underbrace N_{{{52}^{nd}}run}\underbrace A_{{{53}^{rd}}run}\underbrace N_{{{54}^{th}}run}\underbrace {AAA}_{{{55}^{th}}run}\)
\(\underbrace A_{{{56}^{th}}run}\underbrace N_{{{57}^{th}}run}\underbrace {AA}_{{{58}^{th}}run}\underbrace N_{{{59}^{th}}run}\underbrace A_{{{60}^{th}}run}\underbrace N_{{{61}^{st}}run}\underbrace A_{{{62}^{nd}}run}\underbrace {NNN}_{{{63}^{rd}}run}\underbrace A_{{{64}^{th}}run}\underbrace N_{{{65}^{th}}run}\)
The total number of runs is denoted by G is equal to 6.
Here,\({n_1} > 20\)and\({n_2} > 20\). The value of the test statistic z needs to be calculated.
The mean value of G is calculated as follows:
\(\begin{array}{c}{\mu _G} = \frac{{2{n_1}{n_2}}}{{{n_1} + {n_2}}} + 1\\ = \frac{{2\left( {47} \right)\left( {63} \right)}}{{47 + 63}} + 1\\ = 54.84\end{array}\)
The standard deviation of G is computed as follows:
\(\begin{array}{c}{\sigma _G} = \sqrt {\frac{{2{n_1}{n_2}\left( {2{n_1}{n_2} - {n_1} - {n_2}} \right)}}{{{{\left( {{n_1} + {n_2}} \right)}^2}\left( {{n_1} + {n_2} - 1} \right)}}} \\ = \sqrt {\frac{{2\left( {47} \right)\left( {63} \right)\left( {2\left( {47} \right)\left( {63} \right) - 47 - 63} \right)}}{{{{\left( {47 + 63} \right)}^2}\left( {47 + 63 - 1} \right)}}} \\ = 5.108\end{array}\)
Thus, the test statistic (G) is computed below:
\(\begin{array}{c}z = \frac{{G - {\mu _G}}}{{{\sigma _G}}}\\ = \frac{{65 - 54.84}}{{5.108}}\\ = 1.989\end{array}\)
