To solve this problem, we will conduct a hypothesis test for proportions. We are given the following:

• Historically: $p_0 = 0.5858$ (historical proportion of people who believe in global warming).
• Sample size: $n = 978$ (number of adults in the poll).
• Number of people who said "yes": $x = 530$ (number of people in the sample who believe global warming is real).
• Sample proportion: $\hat{p} = \frac{x}{n} = \frac{530}{978}$.

We are to perform a hypothesis test to determine if the proportion of people who believe in global warming has changed from the historical value of 58.58%. We will complete parts a through c.

a) State the hypotheses

We are conducting a two-tailed test since we are trying to determine if the proportion has changed, not just decreased or increased. Therefore, our hypotheses are:

• Null Hypothesis ($H_0$): The proportion of people who believe in global warming has not changed, $p = 0.5858$.
• Alternative Hypothesis ($H_a$): The proportion of people who believe in global warming has changed, $p \neq 0.5858$.

b) Calculate the sample proportion

The sample proportion $\hat{p}$ is calculated as:

$\hat{p} = \frac{x}{n} = \frac{530}{978} \approx 0.5419$

c) Determine the $z$-test statistic

The formula for the $z$-test statistic in a proportion hypothesis test is:

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$

Now, we will calculate this step by step.

1. Calculate the standard error:

$\text{Standard error} = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.5858(1 - 0.5858)}{978}}$

2. Calculate the test statistic:

$z = \frac{0.5419 - 0.5858}{\text{Standard error}}$

I'll calculate the value of the $z$-test statistic.The $z$-test statistic is approximately $-2.79$.

This means that the observed sample proportion is 2.79 standard errors below the historical proportion. To make a conclusion, we can compare this $z$-value with critical values for a two-tailed test at a given significance level (e.g., 0.05).

Would you like to continue by finding the p-value and drawing a conclusion from this hypothesis test?

Here are some related questions:

1. What is the significance level in a hypothesis test?
2. How do you interpret a p-value in a z-test for proportions?
3. Why do we use the normal distribution in large sample size hypothesis tests for proportions?
4. How would the result change if the sample size were smaller?
5. What could be the implications of a change in the proportion of people believing in global warming?

Tip: The further away the test statistic is from 0, the more evidence there is against the null hypothesis.