To compute the value of the test statistic, we will use the z-test for a population proportion since we are testing a proportion against a hypothesized value. The formula for the test statistic is:

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$

Step-by-Step Calculation:

1. Define the variables:

   • Sample proportion ($\hat{p}$):
     $\hat{p} = \frac{\text{number of procrastinators in the sample}}{\text{sample size}} = \frac{254}{277} \approx 0.9162$
   • Hypothesized population proportion ($p_0$): $0.94$
   • Sample size ($n$): $277$

2. Calculate the standard error: $\text{SE} = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.94(1 - 0.94)}{277}} = \sqrt{\frac{0.94 \times 0.06}{277}} \approx \sqrt{0.00020385} \approx 0.01429$

3. Compute the test statistic: $z = \frac{\hat{p} - p_0}{\text{SE}} = \frac{0.9162 - 0.94}{0.01429} \approx \frac{-0.0238}{0.01429} \approx -1.67$

Final Answer:

The value of the test statistic is approximately: $z = -1.67$

Let me know if you have any questions or would like to proceed with further steps in this hypothesis test!

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Here are 5 questions to deepen your understanding:

1. What is the significance of the hypothesized proportion ($p_0$) in this context?
2. Why do we use a z-test instead of a t-test for proportions?
3. How does the sample size affect the standard error and test statistic?
4. What role does the significance level ($\alpha = 0.10$) play in interpreting the results?
5. What conclusion would you draw if the calculated $z$-value fell inside the critical region?

Tip: Always check assumptions of the z-test (large sample size and independent observations) before applying it.