QUESTION 5 Historically, the proportion of students entering a university who finished in 4 years or less was \( 63 \% \). To test whether this proportion has decreased, 114 students were examined and \( 51 \% \) had finished in 4 years or less. To determine whether the proportion of students who finish in 4 year or less has statistically significantly decreased (at the \( 5 \% \) level of signficance), what is the critical value? (please round your answer to 2 decimal places)

To determine the critical value for testing whether the proportion of students who finish in 4 years or less has **decreased** from the historical proportion of \( 63\% \), follow these steps: ### Step 1: Define the Hypotheses - **Null Hypothesis (\( H_0 \))**: \( p \geq 0.63 \) (The proportion has not decreased) - **Alternative Hypothesis (\( H_a \))**: \( p < 0.63 \) (The proportion has decreased) ### Step 2: Determine the Significance Level - The significance level (\( \alpha \)) is \( 5\% \) or \( 0.05 \). ### Step 3: Identify the Critical Value - Since this is a **left-tailed test**, you need to find the z-score that corresponds to the cumulative probability of \( \alpha = 0.05 \). Using standard normal distribution tables or a calculator: - The critical z-value for a left-tailed test at \( \alpha = 0.05 \) is approximately **-1.645**. ### Step 4: Round the Critical Value - Rounded to two decimal places, the critical value is **-1.65**. ### Conclusion If the calculated test statistic is less than **-1.65**, you would reject the null hypothesis, concluding that there is significant evidence at the \( 5\% \) level to suggest that the proportion of students finishing in 4 years or less has decreased. **Critical Value:** \(-1.65\)