Questions: In a Gallup poll about the cause of record highs in 2015 temperatures, 49% of adults answered "human-caused climate change," 46% chose "natural changes in the Earth's temperatures," and 5% had no opinion. Suppose that in a random sample of adults this year, 55% attribute warmer weather patterns to "human-caused climate change." Is the percentage of the public with this opinion higher this year than in 2015? We test the following hypotheses: Ho: p=0.49, Ha: p>0.49. The P-value is 0.03. Interpret the P-value as a probability statement.

In a Gallup poll about the cause of record highs in 2015 temperatures, 49% of adults answered "human-caused climate change," 46% chose "natural changes in the Earth's temperatures," and 5% had no opinion.

Suppose that in a random sample of adults this year, 55% attribute warmer weather patterns to "human-caused climate change."

Is the percentage of the public with this opinion higher this year than in 2015? We test the following hypotheses: Ho: p=0.49, Ha: p>0.49. The P-value is 0.03. Interpret the P-value as a probability statement.

Transcript text: 4) In a Gallup poll about the cause of record highs in 2015 temperatures, $49 \%$ of adults answered "human-caused climate change," $46 \%$ chose "natural changes in the Earth's temperatures," and $5 \%$ had no opinion. Suppose that in a random sample of adults this year, $55 \%$ attribute warmer weather patterns to "human-caused climate change." Is the percentage of the public with this opinion higher this year than in 2015? We test the following hypotheses: Ho: $\mathrm{p}=0.49, \mathrm{Ha}: \mathrm{p}>0.49$. The P -value is 0.03 . Interpret the P -value as a probability statement. Page 179

Solution

Solution Steps

Step 1: Hypothesis Formulation

We are testing the following hypotheses:

Null Hypothesis ($H_0$): $p = 0.49$ (the proportion of adults attributing warmer weather patterns to human-caused climate change is the same as in 2015)
Alternative Hypothesis ($H_a$): $p > 0.49$ (the proportion is higher this year than in 2015)

Step 2: Test Statistic Calculation

The test statistic $Z$ is calculated using the formula: \[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Substituting the values:

$\hat{p} = 0.55$ (sample proportion this year)
$p_0 = 0.49$ (hypothesized proportion)
$n = 1000$ (sample size)

Calculating: \[ Z = \frac{0.55 - 0.49}{\sqrt{\frac{0.49(1 - 0.49)}{1000}}} = 3.7955 \]

Step 3: P-value Calculation

The P-value associated with the test statistic $Z = 3.7955$ is found to be: \[ \text{P-value} = 0.0001 \]

Step 4: Critical Region and Decision

The critical value for a one-tailed test at the significance level $\alpha = 0.05$ is: \[ Z_{critical} = 1.6449 \] Since $Z = 3.7955 > 1.6449$, we are in the critical region.

Step 5: Conclusion

The P-value $0.0001$ is less than the significance level $\alpha = 0.05$. Therefore, we reject the null hypothesis $H_0$.

This suggests that the percentage of the public with the opinion that warmer weather patterns are due to human-caused climate change is significantly higher this year than in 2015.