This problem requires a hypothesis test for the difference in proportions. Let’s solve step-by-step:

1. Define Hypotheses

Let $p_1$ be the proportion of students planning to study health-related fields now and $p_2$ the proportion 10 years ago.

• Null Hypothesis ($H_0$): $p_1 - p_2 \leq 0$ (The proportion now is less than or equal to 10 years ago.)
• Alternative Hypothesis ($H_a$): $p_1 - p_2 > 0$ (The proportion now is greater than 10 years ago.)

This is a right-tailed test.

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2. Gather the Data

• Group 1: $n_1 = 400$, $\hat{p}_1 = 0.18$
• Group 2: $n_2 = 600$, $\hat{p}_2 = 0.15$

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3. Compute the Pooled Proportion

The pooled proportion is given by: $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$ where: $x_1 = \hat{p}_1 \cdot n_1 = 0.18 \cdot 400 = 72$ $x_2 = \hat{p}_2 \cdot n_2 = 0.15 \cdot 600 = 90$

$\hat{p} = \frac{72 + 90}{400 + 600} = \frac{162}{1000} = 0.162$

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4. Compute the Test Statistic

The test statistic $z$ is: $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$

Substituting the values: $z = \frac{0.18 - 0.15}{\sqrt{0.162 \cdot (1 - 0.162) \cdot \left(\frac{1}{400} + \frac{1}{600}\right)}}$ $z = \frac{0.03}{\sqrt{0.162 \cdot 0.838 \cdot \left(0.0025 + 0.001667\right)}}$ $z = \frac{0.03}{\sqrt{0.162 \cdot 0.838 \cdot 0.004167}}$ $z = \frac{0.03}{\sqrt{0.000566}}$ $z = \frac{0.03}{0.0238} \approx 1.2605$

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5. Compute the p-value

Using a standard normal table or calculator, for $z = 1.2605$, the p-value for a one-tailed test is: $\text{p-value} \approx 0.1039$

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6. Decision

At a 1% significance level ($\alpha = 0.01$), the p-value ($0.1039$) is greater than $0.01$. Therefore, we fail to reject the null hypothesis. There is not enough evidence to conclude that the proportion of students planning to study health-related fields is greater now than 10 years ago.

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Final Answers

• Alternative Sign: $>$ (right-tailed test)
• Test Statistic: $z \approx 1.2605$
• p-value: $\approx 0.1039$

Would you like a more detailed explanation of any step?

Related Questions:

1. What is the importance of pooling proportions in hypothesis tests?
2. How do sample sizes affect the test statistic in this problem?
3. Why is this a right-tailed test instead of a two-tailed test?
4. How does significance level influence decision-making in hypothesis tests?
5. What other methods could be used to compare proportions in such surveys?

Tip: Always check if the assumptions for a hypothesis test (e.g., large enough sample size for normal approximation) are satisfied before proceeding!