​Historically, about

5858​%

of the population of a certain country believed that the​ planet's temperature was rising​ ("global warming"). A March 2010 poll wanted to determine whether this proportion had changed. The poll interviewed

978978

adults in the​ population, and

530530

said they believed that global warming was real.​ (Assume these

978978

adults represented a simple random​ sample. ​Historically, about

5858​%

of the population of a certain country believed that the​ planet's temperature was rising​ ("global warming"). A March 2010 poll wanted to determine whether this proportion had changed. The poll interviewed

978978

adults in the​ population, and

530530

said they believed that global warming was real.​ (Assume these

978978

adults represented a simple random​ sample.) Complete parts a through c below. ​Historically, about 

5858​% 

of the population of a certain country believed that the​ planet's temperature was rising​ ("global warming"). A March 2010 poll wanted to determine whether this proportion had changed. The poll interviewed 

978978 

adults in the​ population, and 

530530 

said they believed that global warming was real.​ (Assume these 

978978 

adults represented a simple random​ sample.) Complete parts a through c below.Determine the​ z-test statistic, using the value for 

ModifyingAbove p with caretp 

from part a. 


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.

Historically, about 58.58% of the population of a certain country believed that the planet's temperature was rising ('global warming'). A March 2010 poll wanted to determine whether this proportion had changed. The poll interviewed 978 adults in the population, and 530 said they believed that global warming was real. (Assume these 978 adults represented a simple random sample.) Complete parts a through c. Determine the z-test statistic, using the value for p from part a.

Explore a hypothesis test to determine if the proportion of people believing in global warming has changed. The problem uses a z-test for proportions with a historical value of 58.58%, a sample size of 978, and 530 respondents. Learn how to compute the sample proportion, z-test statistic, and draw conclusions from the test. Suitable for undergraduate statistics students.

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