1. **Problem statement:** We want to test the hypothesis that gender (Køn) and sport participation (Sport) are independent. 2. **Hypotheses:** - Null hypothesis $H_0$: Gender and sport are independent. - Alternative hypothesis $H_1$: Gender and sport are dependent. 3. **Data from the table:** \begin{align*} &\text{Sport Yes (Ja)}: \text{Boys (Dreng)} = 1590, \text{Girls (Pige)} = 1451, \text{Total} = 3041 \\ &\text{Sport No (Nej)}: \text{Boys} = 116, \text{Girls} = 146, \text{Total} = 262 \\ &\text{Total Boys} = 1706, \text{Total Girls} = 1597, \text{Grand Total} = 3303 \end{align*} 4. **Test used:** Chi-square test for independence. 5. **Expected counts under $H_0$:** $$ E_{ij} = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}} $$ Calculate expected counts: - $E_{\text{Ja, Dreng}} = \frac{3041 \times 1706}{3303} = \frac{5189246}{3303} \approx 1570.5$ - $E_{\text{Ja, Pige}} = \frac{3041 \times 1597}{3303} = \frac{4854977}{3303} \approx 1470.5$ - $E_{\text{Nej, Dreng}} = \frac{262 \times 1706}{3303} = \frac{447572}{3303} \approx 135.5$ - $E_{\text{Nej, Pige}} = \frac{262 \times 1597}{3303} = \frac{418114}{3303} \approx 126.5$ 6. **Calculate Chi-square statistic:** $$ \chi^2 = \sum \frac{(O - E)^2}{E} $$ Where $O$ is observed count and $E$ is expected count. Calculate each term: - $\frac{(1590 - 1570.5)^2}{1570.5} = \frac{19.5^2}{1570.5} = \frac{380.25}{1570.5} \approx 0.242$ - $\frac{(1451 - 1470.5)^2}{1470.5} = \frac{(-19.5)^2}{1470.5} = \frac{380.25}{1470.5} \approx 0.259$ - $\frac{(116 - 135.5)^2}{135.5} = \frac{(-19.5)^2}{135.5} = \frac{380.25}{135.5} \approx 2.806$ - $\frac{(146 - 126.5)^2}{126.5} = \frac{19.5^2}{126.5} = \frac{380.25}{126.5} \approx 3.005$ Sum: $$ \chi^2 = 0.242 + 0.259 + 2.806 + 3.005 = 6.312 $$ 7. **Degrees of freedom:** $$ df = (\text{rows} - 1)(\text{columns} - 1) = (2-1)(2-1) = 1 $$ 8. **Critical value at 5% significance level:** From chi-square table, $\chi^2_{0.05,1} = 3.841$ 9. **Decision:** Since $6.312 > 3.841$, we reject $H_0$. 10. **Conclusion:** There is sufficient evidence at the 5% significance level to conclude that gender and sport participation are dependent.