Questions: For the following scenario, choose the appropriate set of hypotheses: The graduation rate for students at State College had previously been recorded at 75%. A new administrator states that programs are in place that should increase student graduation rates. H0: μ=0.75 H1: μ>0.75 Ho: p=0.75 H1: p>0.75

For the following scenario, choose the appropriate set of hypotheses: The graduation rate for students at State College had previously been recorded at 75%. A new administrator states that programs are in place that should increase student graduation rates. H0: μ=0.75 H1: μ>0.75 Ho: p=0.75 H1: p>0.75

Transcript text: For the following scenario, choose the appropriate set of hypotheses: The graduation rate for students at State College had previously been recorded at 75%. A new administrator states that programs are in place that should increase student graduation rates. $H_{0}: \mu=0.75 H_{1}: \mu>0.75$ Ho: $p=0.75 \quad \mathrm{H}_{1}: p>0.75$

Solution

Solution Steps

Step 1: State the Hypotheses

We are testing the following hypotheses:

Null Hypothesis: $ H_0: p = 0.75 $
Alternative Hypothesis: $ H_1: p > 0.75 $

Step 2: Calculate the Test Statistic

The test statistic for the proportion is calculated using the formula: \[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Substituting the values:

Sample proportion $ \hat{p} = 0.78 $
Hypothesized proportion $ p_0 = 0.75 $
Sample size $ n = 200 $

The calculated test statistic is: \[ Z = 0.9798 \]

Step 3: Determine the P-value

The P-value associated with the test statistic $ Z = 0.9798 $ is: \[ \text{P-value} = 0.1636 \]

Step 4: Identify the Critical Region

For a significance level of $ \alpha = 0.05 $ in a one-tailed test, the critical value is: \[ Z_{\text{critical}} = 1.6449 \] The critical region is defined as: \[ Z > 1.6449 \]

Step 5: Make a Decision

To make a decision, we compare the test statistic to the critical value: