1. Population, Parameter, Sample, and Statistic

• Population: All American adults.
• Parameter: The true proportion ($p$) of American adults who support the increased usage of coal to produce energy.
• Sample: The 1000 American adults randomly selected by Pew Research.
• Statistic: The sample proportion ($\hat{p}$) of respondents who support increased usage of coal, which is: $\hat{p} = \frac{\text{Number of supporters}}{\text{Sample size}} = \frac{370}{1000} = 0.37$

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2. Hypotheses

To evaluate whether a majority of American adults support the increased usage of coal, we set up hypotheses as follows:

• Null Hypothesis ($H_0$): $p = 0.5$
  (The true proportion of American adults who support increased usage of coal is equal to 50%.)

• Alternative Hypothesis ($H_a$): $p < 0.5$
  (The true proportion of American adults who support increased usage of coal is less than 50%.)

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3. Explanation of the Alternative Hypothesis

The alternative hypothesis ($p < 0.5$) is chosen because the goal is to test whether fewer than half of American adults support increased coal usage. This aligns with the survey result ($\hat{p} = 0.37$), which suggests that the proportion of supporters might indeed be below 50%. Testing for $p < 0.5$ is consistent with evaluating whether the majority opposes increased coal usage.

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Recap

The population represents all American adults, the parameter is the true proportion of supporters ($p$), the sample is 1000 adults surveyed, and the statistic is the sample proportion ($\hat{p} = 0.37$). The hypotheses are framed to evaluate whether fewer than 50% of American adults support increased coal usage.

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Do you have further questions or need assistance with calculating test statistics or interpreting results?
Here are 5 related questions to expand this information:

1. How do you calculate the test statistic and p-value for this hypothesis test?
2. What assumptions need to be checked for this hypothesis test to be valid?
3. How would you interpret the result if the p-value is less than the significance level?
4. What is the margin of error for the sample proportion in this survey?
5. How would the conclusions change if the sample size were larger or smaller?

Tip: Always verify the conditions for statistical inference, such as random sampling and sufficient sample size, to ensure the reliability of hypothesis test results.