We are given a hypothesis test regarding the proportion of people who believe a pregnant woman should be able to obtain a legal abortion under the given conditions. Let's break it down step by step.

(a) Setting up the Hypothesis Test

We define:

• $\pi$ = Population proportion of people who would respond "yes."
• Sample size: $n = 587 + 636 = 1223$
• Sample proportion:
  $\hat{\pi} = \frac{587}{1223} \approx 0.480$

We will conduct three hypothesis tests:

1. Z-test for One Proportion
2. Likelihood Ratio Test
3. Score Test (Wald Test)

We test the null hypothesis:

$H_0: \pi = 0.5 \quad \text{(Null Hypothesis: Population is equally split)}$

against the alternative:

$H_A: \pi \neq 0.5 \quad \text{(Two-tailed test)}$

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(b) Estimating the p-values Using Statistical Tables

1. Z-test for One Proportion (Wald Test)

The test statistic is:

$Z = \frac{\hat{\pi} - 0.5}{\sqrt{\frac{0.5(1-0.5)}{n}}}$

Substituting values:

$Z = \frac{0.480 - 0.5}{\sqrt{\frac{0.25}{1223}}}$

$Z = \frac{-0.02}{\sqrt{0.0002044}}$

$Z = \frac{-0.02}{0.0143} = -1.40$

Using the standard normal table, the p-value for $Z = -1.40$ (two-tailed test) is approximately:

$2 \times P(Z < -1.40) = 2 \times 0.0808 = 0.1616$

So, $p \approx 0.16$.

2. Likelihood Ratio Test

The likelihood ratio statistic is:

$G^2 = 2 \left[ x \ln \left( \frac{x}{n\pi_0} \right) + (n-x) \ln \left( \frac{n-x}{n(1-\pi_0)} \right) \right]$

Substituting values:

$G^2 = 2 \left[ 587 \ln \left( \frac{587}{611.5} \right) + 636 \ln \left( \frac{636}{611.5} \right) \right]$

Approximating calculations:

$G^2 \approx 2 \times (-7.51) = 15.02$

This follows a chi-square distribution with 1 degree of freedom. Using statistical tables, $P(\chi^2 > 15.02) \approx 0.16$, similar to the Z-test.

3. Score Test (Wald Test)

The score test produces a similar statistic to the Z-test, yielding a similar p-value of approximately $0.16$.

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(c) Interpretation of Results

• The p-value is approximately 0.16 for all three tests.
• Since $p > 0.05$, we fail to reject the null hypothesis.
• This means that there is not enough evidence to conclude that the population proportion who support legal abortion in this scenario is significantly different from 50%.
• The data suggest that opinions are fairly balanced, though slightly less than half of respondents support legal abortion in this case.

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Would you like a more detailed breakdown of any test? Let me know!

Further Related Questions

1. How does the sample size affect the power of this hypothesis test?
2. What assumptions are made in the Z-test for a single proportion?
3. How does the likelihood ratio test differ from the Wald test?
4. If the sample proportion were closer to 0.5, how would the p-value change?
5. How can confidence intervals be used to complement hypothesis testing in this scenario?

Tip:

Always check assumptions (such as normality and sample size) before using statistical tests to ensure valid conclusions.