To compute the test statistic and p-value, we'll conduct a hypothesis test for a population proportion.

Step 1: Define Hypotheses

• Null hypothesis ($H_0$): The proportion of American adults who support increased coal usage is $p = 0.5$.
• Alternative hypothesis ($H_a$): The proportion of American adults who support increased coal usage is not $p = 0.5$ (two-tailed test).

Step 2: Sample Statistics

• Sample size ($n$) = 1000
• Observed number of supporters = 370
• Sample proportion ($\hat{p}$) = $\frac{370}{1000} = 0.37$

Step 3: Test Statistic

The test statistic for a proportion follows the formula: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$ where:

• $p_0$ = hypothesized population proportion (0.5)
• $n$ = sample size (1000)
• $\hat{p}$ = observed sample proportion (0.37)

Substitute the values: $z = \frac{0.37 - 0.5}{\sqrt{\frac{0.5(1 - 0.5)}{1000}}}$ $z = \frac{-0.13}{\sqrt{0.00025}} = \frac{-0.13}{0.01581} \approx -8.22$

Step 4: Compute the P-Value

Since this is a two-tailed test, the p-value is: $\text{p-value} = 2 \cdot P(Z < -8.22)$ From the standard normal distribution, the probability for $Z = -8.22$ is effectively 0. Thus: $\text{p-value} \approx 0$

Step 5: Interpretation

With a p-value of approximately 0, we reject the null hypothesis. This indicates that there is strong evidence to conclude that the proportion of Americans who support increased coal usage is significantly different from 50%.

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Would you like more details or assistance with a related problem?
Here are 5 questions to expand understanding:

1. What assumptions are necessary for this hypothesis test to be valid?
2. How would the test statistic change if the sample size were larger?
3. How do we interpret a two-tailed test compared to a one-tailed test?
4. What role does the significance level ($\alpha$) play in deciding whether to reject $H_0$?
5. How does the observed proportion ($\hat{p}$) affect the p-value?

Tip: Always verify if your sample size is large enough for the normal approximation to be valid. Use $np_0 \geq 10$ and $n(1-p_0) \geq 10$ as a rule of thumb.