To solve the problem presented in the image, we need to follow these steps:

1. Calculate the Expected Frequencies:

   • The expected frequency for each category is calculated using the formula: $E_i = n \cdot p_i$ where $n$ is the total number of observations and $p_i$ is the proportion for each category.

   The total number of observations ($n$) is: $n = 19 + 57 + 46 + 12 = 134$

   Using the given proportions $p_A = 0.15$, $p_B = 0.4$, $p_C = 0.3$, $p_D = 0.15$, we calculate:

   $E_A = 134 \cdot 0.15 = 20.1$ $E_B = 134 \cdot 0.4 = 53.6$ $E_C = 134 \cdot 0.3 = 40.2$ $E_D = 134 \cdot 0.15 = 20.1$

2. Compute the Chi-Square Test Statistic:

   • The Chi-Square test statistic is calculated using the formula: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$ where $O_i$ is the observed frequency and $E_i$ is the expected frequency.

   For each category: $\chi^2_A = \frac{(19 - 20.1)^2}{20.1} = \frac{1.21}{20.1} = 0.0602$ $\chi^2_B = \frac{(57 - 53.6)^2}{53.6} = \frac{11.56}{53.6} = 0.2157$ $\chi^2_C = \frac{(46 - 40.2)^2}{40.2} = \frac{33.64}{40.2} = 0.8366$ $\chi^2_D = \frac{(12 - 20.1)^2}{20.1} = \frac{65.61}{20.1} = 3.2632$

   Summing these up: $\chi^2 = 0.0602 + 0.2157 + 0.8366 + 3.2632 = 4.3757$

   Rounded to three decimal places, $\chi^2 \approx 4.376$.

3. Determine the p-value:

   • To find the p-value, we use the Chi-Square distribution with $df = k - 1$ degrees of freedom, where $k$ is the number of categories. $df = 4 - 1 = 3$

   Using