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Sagot :
Let me walk you through the process of finding the expected values in a Chi-Square test for independence based on the provided contingency table.
First, we need to collect the totals from our contingency table:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{CDs} & \text{No CDs} & \text{Row Total} \\ \hline \text{Smartphone} & 5 & 25 & 30 \\ \hline \text{No Smartphone} & 17 & 20 & 37 \\ \hline \text{Column Total} & 22 & 45 & 67 \\ \hline \end{array} \][/tex]
Next, we use the formula for expected values in each cell:
[tex]\[ E = \frac{{(\text{Row Total}) \times (\text{Column Total})}}{{\text{Grand Total}}} \][/tex]
Now, let’s calculate each expected value step by step:
1. Expected value for Smartphones & CDs:
[tex]\[ E(\text{Smartphone, CDs}) = \frac{{\text{Row Total (Smartphones)} \times \text{Column Total (CDs)}}}{{\text{Grand Total}}} = \frac{{30 \times 22}}{67} \approx 9.85 \][/tex]
2. Expected value for Smartphones & No CDs:
[tex]\[ E(\text{Smartphone, No CDs}) = \frac{{\text{Row Total (Smartphones)} \times \text{Column Total (No CDs)}}}{{\text{Grand Total}}} = \frac{{30 \times 45}}{67} \approx 20.15 \][/tex]
3. Expected value for No Smartphones & CDs:
[tex]\[ E(\text{No Smartphone, CDs}) = \frac{{\text{Row Total (No Smartphones)} \times \text{Column Total (CDs)}}}{{\text{Grand Total}}} = \frac{{37 \times 22}}{67} \approx 12.15 \][/tex]
4. Expected value for No Smartphones & No CDs:
[tex]\[ E(\text{No Smartphone, No CDs}) = \frac{{\text{Row Total (No Smartphones)} \times \text{Column Total (No CDs)}}}{{\text{Grand Total}}} = \frac{{37 \times 45}}{67} \approx 24.85 \][/tex]
Placing these expected values in our table, we get:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{CDs} & \text{No CDs} & \text{Row Total} \\ \hline \text{Smartphone} & 5\ (9.85) & 25\ (20.15) & 30 \\ \hline \text{No Smartphone} & 17\ (12.15) & 20\ (24.85) & 37 \\ \hline \text{Column Total} & 22 & 45 & 67 \\ \hline \end{array} \][/tex]
Using the above table, we identify the correct answer from the given options is:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline \multirow{4}{*}{ O } & & \text{CDs} & \text{No CDs} & \text{Row Total} & \\ \hline & \text{Smartphone} & 5\ (9.85) & 25\ (20.15) & 30 & \\ \hline & \text{No Smartphone} & 17\ (12.15) & 20\ (24.85) & 37 & \\ \hline & \text{Column Total} & 22 & 45 & 67 & \\ \hline \end{array} \][/tex]
First, we need to collect the totals from our contingency table:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{CDs} & \text{No CDs} & \text{Row Total} \\ \hline \text{Smartphone} & 5 & 25 & 30 \\ \hline \text{No Smartphone} & 17 & 20 & 37 \\ \hline \text{Column Total} & 22 & 45 & 67 \\ \hline \end{array} \][/tex]
Next, we use the formula for expected values in each cell:
[tex]\[ E = \frac{{(\text{Row Total}) \times (\text{Column Total})}}{{\text{Grand Total}}} \][/tex]
Now, let’s calculate each expected value step by step:
1. Expected value for Smartphones & CDs:
[tex]\[ E(\text{Smartphone, CDs}) = \frac{{\text{Row Total (Smartphones)} \times \text{Column Total (CDs)}}}{{\text{Grand Total}}} = \frac{{30 \times 22}}{67} \approx 9.85 \][/tex]
2. Expected value for Smartphones & No CDs:
[tex]\[ E(\text{Smartphone, No CDs}) = \frac{{\text{Row Total (Smartphones)} \times \text{Column Total (No CDs)}}}{{\text{Grand Total}}} = \frac{{30 \times 45}}{67} \approx 20.15 \][/tex]
3. Expected value for No Smartphones & CDs:
[tex]\[ E(\text{No Smartphone, CDs}) = \frac{{\text{Row Total (No Smartphones)} \times \text{Column Total (CDs)}}}{{\text{Grand Total}}} = \frac{{37 \times 22}}{67} \approx 12.15 \][/tex]
4. Expected value for No Smartphones & No CDs:
[tex]\[ E(\text{No Smartphone, No CDs}) = \frac{{\text{Row Total (No Smartphones)} \times \text{Column Total (No CDs)}}}{{\text{Grand Total}}} = \frac{{37 \times 45}}{67} \approx 24.85 \][/tex]
Placing these expected values in our table, we get:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{CDs} & \text{No CDs} & \text{Row Total} \\ \hline \text{Smartphone} & 5\ (9.85) & 25\ (20.15) & 30 \\ \hline \text{No Smartphone} & 17\ (12.15) & 20\ (24.85) & 37 \\ \hline \text{Column Total} & 22 & 45 & 67 \\ \hline \end{array} \][/tex]
Using the above table, we identify the correct answer from the given options is:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline \multirow{4}{*}{ O } & & \text{CDs} & \text{No CDs} & \text{Row Total} & \\ \hline & \text{Smartphone} & 5\ (9.85) & 25\ (20.15) & 30 & \\ \hline & \text{No Smartphone} & 17\ (12.15) & 20\ (24.85) & 37 & \\ \hline & \text{Column Total} & 22 & 45 & 67 & \\ \hline \end{array} \][/tex]
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