To determine if the percentage of students who smoke has changed over the last five years, we need to conduct a hypothesis test for the population proportion. This involves comparing the observed sample proportion with the historical population proportion (from five years ago) and calculating a standardized test statistic. Here's the step-by-step process:

Step 1: Define the Hypotheses

• Null Hypothesis (H₀): The percentage of students who smoke has not changed. $H_0: p = 0.18$
• Alternative Hypothesis (H₁): The percentage of students who smoke has changed. $H_1: p \neq 0.18$ This will be a two-tailed test since we're checking for any change (increase or decrease) in the proportion of smokers.

Step 2: Calculate the Sample Proportion

From the sample of 200 students, 44 of them smoke. The sample proportion ($\hat{p}$) is: $\hat{p} = \frac{44}{200} = 0.22$

Step 3: Calculate the Standard Error (SE)

The standard error for the proportion is calculated using the formula: $SE = \sqrt{\frac{p(1 - p)}{n}}$ where $p = 0.18$ (the population proportion from five years ago) and $n = 200$ (the sample size).

$SE = \sqrt{\frac{0.18(1 - 0.18)}{200}} = \sqrt{\frac{0.18 \times 0.82}{200}} = \sqrt{\frac{0.1476}{200}} = \sqrt{0.000738} \approx 0.0272$

Step 4: Calculate the Test Statistic (z)

The standardized test statistic (z) is calculated as: $z = \frac{\hat{p} - p}{SE}$ Substituting the values: $z = \frac{0.22 - 0.18}{0.0272} = \frac{0.04}{0.0272} \approx 1.471$

Conclusion:

The standardized test statistic is approximately 1.471 (rounded to 3 decimal places).

Would you like further details or clarifications on any step?

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Here are 5 related questions to explore:

1. What is the p-value associated with a z-score of 1.471 in a two-tailed test?
2. How would the conclusion of the hypothesis test change if the sample size were larger?
3. What does a Type I error mean in the context of this hypothesis test?
4. What is the critical value for a 5% significance level in a two-tailed z-test?
5. How does changing the significance level affect the outcome of the hypothesis test?

Tip: The standard error decreases as the sample size increases, making the test more sensitive to small differences.