Math Solutions

(All Steps) Age versus Treatment The following table shows the corresponding contingency table: Observed Age < 65 Age 65+ Total New Medication 40 20 60

Question: Age versus Treatment

The following table shows the corresponding contingency table:

Observed Age < 65 Age 65+ Total
New Medication 40 20 60
Standard Medication 25 35 60
Total 65 55 120

We are interested in testing the following null and alternative hypotheses:

\[\begin{aligned}{{H}_{0}}:\,\,\, \text{Treatment}\text{ and }\text {Age}\text{ are independent} \\ {{H}_{A}}:\,\,\,\text{Treatment}\text{ and }\text {Age}\text{ are NOT independent} \\ \end{aligned}\]

From the table above we compute the table with the expected values

Expected Age < 65 Age 65+
New Medication 32.5 27.5
Standard Medication 32.5 27.5

The way those expected frequencies are calculated is shown below:

\[{E}_{{1},{1}}= \frac{ {R}_{1} \times {C}_{1} }{T}= \frac{{60} \times {65}}{{120}}={32.5},\,\,\,\, {E}_{{1},{2}}= \frac{ {R}_{1} \times {C}_{2} }{T}= \frac{{60} \times {55}}{{120}}={27.5},\,\,\,\, {E}_{{2},{1}}= \frac{ {R}_{2} \times {C}_{1} }{T}= \frac{{60} \times {65}}{{120}}={32.5}\] \[,\,\,\,\, {E}_{{2},{2}}= \frac{ {R}_{2} \times {C}_{2} }{T}= \frac{{60} \times {55}}{{120}}={27.5}\]

Finally, we use the formula \(\frac{{{\left( O-E \right)}^{2}}}{E}\) to get

(fo - fe)²/fe Age < 65 Age 65+
New Medication 1.7308 2.0455
Standard Medication 1.7308 2.0455

The calculations required are shown below:

\[\frac{ {\left( {40}-{32.5} \right)}^{2} }{{32.5}} ={1.7308},\,\,\,\, \frac{ {\left( {20}-{27.5} \right)}^{2} }{{27.5}} ={2.0455},\,\,\,\, \frac{ {\left( {25}-{32.5} \right)}^{2} }{{32.5}} ={1.7308},\,\,\,\, \frac{ {\left( {35}-{27.5} \right)}^{2} }{{27.5}} ={2.0455}\]

Hence, the value of Chi-Square statistics is

\[{{\chi }^{2}}=\sum{\frac{{{\left( {{O}_{ij}}-{{E}_{ij}} \right)}^{2}}}{{{E}_{ij}}}}={1.7308} + {2.0455} + {1.7308} + {2.0455} = 7.552\]

The critical Chi-Square value for \(\alpha =0.05\) and \(\left( 2-1 \right)\times \left( 2-1 \right)=1\) degrees of freedom is \(\chi _{C}^{2}= {3.841}\). Since \({{\chi }^{2}}=\sum{\frac{{{\left( {{O}_{ij}}-{{E}_{ij}} \right)}^{2}}}{{{E}_{ij}}}}= {7.552}\) > \(\chi _{C}^{2}= {3.841}\), then we reject the null hypothesis, which means that we have enough evidence to reject the null hypothesis of independence.

Hence, AGE is related to the treatment .

Age versus Pain Score

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