To fill out the table with the expected counts based on the provided hypotheses for the chi-square goodness-of-fit test, follow these steps:
1. Identify the hypotheses and sample data:
- Null hypothesis: [tex]\( H_0: p_A = 0.5, p_B = 0.25, p_C = 0.25 \)[/tex]
- Total sample count: 391
2. Calculate the expected counts:
- Expected count for category [tex]\( A \)[/tex]: [tex]\( p_A \times \text{Total} \)[/tex]
- Expected count for category [tex]\( B \)[/tex]: [tex]\( p_B \times \text{Total} \)[/tex]
- Expected count for category [tex]\( C \)[/tex]: [tex]\( p_C \times \text{Total} \)[/tex]
3. Substitute the given probabilities and total into the formulas:
- Expected count for [tex]\( A \)[/tex]: [tex]\( 0.5 \times 391 = 195.5 \)[/tex]
- Expected count for [tex]\( B \)[/tex]: [tex]\( 0.25 \times 391 = 97.75 \)[/tex]
- Expected count for [tex]\( C \)[/tex]: [tex]\( 0.25 \times 391 = 97.75 \)[/tex]
4. Fill in the table with the calculated expected counts:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
\text{Category:} & A & B & C \\
\hline
\begin{array}{c}
\text{Expected} \\
\text{count:}
\end{array} & 195.5 & 97.75 & 97.75 \\
\hline
\end{array}
\][/tex]
So the expected counts for each category are:
- [tex]\( \text{Category A: } 195.5 \)[/tex]
- [tex]\( \text{Category B: } 97.75 \)[/tex]
- [tex]\( \text{Category C: } 97.75 \)[/tex]
