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Chapter 8: Problem 37

Historically (from about 2001 to 2014 ), \(57 \%\) of Americans believed that global warming is caused by human activities. A March 2017 Gallup poll of a random sample of 1018 Americans found that 692 believed that global warming is caused by human activities. a. What percentage of the sample believed global warming was caused by human activities? b. Test the hypothesis that the proportion of Americans who believe global warming is caused by human activities has changed from the historical value of \(57 \%\). Use a significance level of \(0.01\). c. Choose the correct interpretation: i. In 2017 , the percentage of Americans who believe global warming is caused by human activities is not significantly different from \(57 \%\). ii. In 2017 , the percentage of Americans who believe global warming is caused by human activities has changed from the historical level of \(57 \%\).

Short Answer

Expert verified
The short answer will be derived from the results of the statistical test and the chosen interpretation. We cannot provide a specific short answer without the actual calculations and results.

Step by step solution

01

Calculate sample percentage

To calculate the percentage of the sample that believed global warming is caused by human activities, divide the number of people who believe (692) by the total number of people in the sample (1018) and then multiply by 100. This will give us the percentage.
02

Set up the hypothesis test

We are testing the hypothesis that the proportion has changed from the historical value of 57%. So, set up the null hypothesis (\(H_0\)) that the proportion of people who believe global warming is caused by human activities, \(p\), is 0.57 (as a decimal). The alternative hypothesis (\(H_1\)) is that \(p\) is different from 0.57.
03

Calculate the test statistic

We use the formula for a z-score in a one sample test of proportion. The test statistic \(z\) is calculated by subtracting the hypothesized proportion (\(p_0\)) from the sample proportion (\(p\)), and dividing by the standard error of the sample proportion. The standard error is calculated as \(\sqrt{p_0(1-p_0)/n}\), where \(n\) is the sample size.
04

Find the p-value

The p-value is the probability of observing a statistic at least as extreme as our test statistic, assuming the null hypothesis is true. We can find this probability using a normal distribution table or a z-table.
05

Make a conclusion

If the p-value is less than our significance level (0.01), we reject the null hypothesis. This means we have evidence to suggest the proportion has changed from 57% (the historical value). If the p-value is greater than our significance level, we fail to reject the null hypothesis. This means we do not have sufficient evidence to suggest that the proportion has changed from 57%.
06

Choose the correct interpretation

Based on our conclusion from the hypothesis test, we choose the corresponding interpretation. If we rejected the null hypothesis, then we would choose interpretation ii. If we did not reject the null hypothesis, then we would choose interpretation i.

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Most popular questions from this chapter

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