Question

Let 𝜋 be the proportion of a population vaccinated against a disease. An estimate ̂𝜋 = 0.64 is found using a sample of 100 individuals from the population. The 𝑧 test statistic for the null hypothesis 𝐻₀ : 𝜋 = 0.58 is ________ (round off to 2 decimal places).

Answer 1

Correct Answer: 1.2

We use the one–sample proportion z-test statistic: 

\[ z = \frac{\hat{\pi}-\pi_0}{\sqrt{\pi_0(1-\pi_0)/n}} \] 
Given:
\[ \hat{\pi} = 0.64,\quad \pi_0 = 0.58,\quad n = 100. \]
 Step 1 — Compute standard error:
\[ SE = \sqrt{\frac{0.58(1-0.58)}{100}} \] \[ = \sqrt{\frac{0.58 \cdot 0.42}{100}} \] \[ = \sqrt{\frac{0.2436}{100}} \] \[ = \sqrt{0.002436} = 0.04935. \] 
Step 2 — Compute test statistic:
\[ z = \frac{0.64 - 0.58}{0.04935} \] \[ = \frac{0.06}{0.04935} \] \[ = 1.216 \approx 1.20. \] 
Final Answer: 1.20