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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

Given the 4 categories and distribution, apply the Goodness-of-fit test for the claim that the 4 categories occur with the following distribution: H0: pA = 0.15, pB = 0.4, pC = 0.3, pD = 0.15. Observed frequencies: A = 19, B = 57, C = 46, D = 12. Complete the table for the expected frequencies, find the chi-square test statistic, p-value, and determine if you reject or fail to reject the null hypothesis at a 0.025 significance level.

Learn how to apply the Chi-Square Goodness-of-Fit test to four categories with observed frequencies of 19, 57, 46, and 12, using expected proportions of 0.15, 0.4, 0.3, and 0.15. This problem demonstrates how to calculate expected frequencies, the chi-square test statistic, and p-value, as well as make decisions based on a significance level of 0.025.

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