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Offshore drilling, Part II. Results of a poll evaluating support for drilling for oil and natural gas off the coast of California were introduced in Exercise \(3.29 .\) \begin{tabular}{lcc} & \multicolumn{2}{c} { College Grad } \\ \cline { 2 - 3 } & Yes & No \\ \hline Support & 154 & 132 \\ Oppose & 180 & 126 \\ Do not know & 104 & 131 \\ \hline Total & 438 & 389 \end{tabular} (a) What percent of college graduates and what percent of the non-college graduates in this sample support drilling for oil and natural gas off the Coast of California? (b) Conduet a hypothesis test to determine if the data provide strong evidence that the proportion of college graduates who support off-shore drilling in California is different than that of noncollege graduates.

Step by step solution

01

Calculate Percentages for College Graduates

To find the percentage of college graduates who support drilling, divide the number of college grads who support drilling by the total number of college grads: \(\frac{154}{438} \times 100\). This gives approximately \(35.16\%\).

02

Calculate Percentages for Non-College Graduates

To find the percentage of non-college graduates who support drilling, divide the number of non-college grads who support drilling by the total number of non-college grads: \(\frac{132}{389} \times 100\). This gives approximately \(33.93\%\).

03

Formulate Hypotheses for the Test

To determine if there is a difference in proportions, set up your null hypothesis (\(H_0\)): The proportion of college graduates who support drilling is equal to that of non-college graduates, \(p_1 = p_2\). The alternative hypothesis (\(H_a\)): The proportions are different, \(p_1 eq p_2\).

04

Calculate Test Statistic

The test statistic for comparing two proportions can be calculated using the formula: \[Z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}\] where \(\hat{p}_1\) and \(\hat{p}_2\) are the sample proportions and \(\hat{p}\) is the pooled sample proportion: \[\hat{p} = \frac{154+132}{438+389}\]Calculate the pooled proportion and then use it to compute the test statistic.

05

Determine Critical Value and Compare

The critical value from the standard normal distribution for a significance level \(\alpha = 0.05\) is approximately \(Z = \pm1.96\). If the calculated \(Z\)-value exceeds the critical value, reject the null hypothesis. Solve for \(Z\) and compare to decide.

06

Calculate Conclusion of Hypothesis Test

Compute the \(Z\) value using the test statistic formula derived in Step 4. Compare this value with the critical \(Z\) value. If the absolute value of the calculated \(Z\) is greater than 1.96, there is strong evidence to reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion Comparison

Understanding how to compare proportions is key in this type of analysis. A common scenario is comparing the proportion of a specific outcome, like support for offshore drilling, between two different groups. In our case, let's examine college graduates versus non-college graduates.
Here's how to approach it:

• Start by determining the proportion of respondents who support drilling within each group.
• For college graduates, the proportion is calculated by dividing those who support by the total number surveyed. Here, divide 154 by 438 and multiply by 100 to get approximately 35.16%.
• Next, do the same for non-college graduates: 132 supporting out of 389 surveyed, which is around 33.93%.

When comparing these proportions, you're assessing whether or not the difference between them is statistically significant. It's important to differentiate between a numeric difference, which could be due to chance, and a difference that is statistically supported by the data.

Statistical Analysis

Hypothesis testing involves making inferences about populations based on sample data. In our case, the hypothesis test aims to identify if a real difference exists between the support levels from college and non-college graduates.
Here's a step-by-step breakdown:

• Begin by setting up your hypotheses. The null hypothesis (\(H_0\)) posits no difference between the groups' proportions, whereas the alternative hypothesis (\(H_a\)) suggests a difference exists.
• Next, calculate the test statistic using the Z-test formula for comparing two proportions. Factors here include the sample proportions and pooled proportion.
• The pooled proportion is a weighted average of both groups: \(\hat{p} = \frac{154+132}{438+389}\).
• Plug these values into the formula: \[Z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}\]
• This calculation provides the Z-score, which measures how many standard deviations the observed difference is from the null hypothesis expectation.

Finally, compare the computed Z-score with the critical value from the standard normal distribution, typically \(±1.96\) for a 5% significance level. If the Z-score exceeds the critical value, you conclude there's strong evidence supporting a real difference.

Offshore Drilling Support

Support for offshore drilling is often a polarizing issue in communities, such as those in California. This particular analysis sheds light on how education level might influence public opinion on energy and environmental policies.
Key considerations include:

• Understanding that public support for or against drilling can have tremendous socio-economic and environmental implications.
• Education often shapes perspectives about complex issues, reflecting varying awareness levels about environmental impacts and resource management.
• Differences highlighted by our analysis can lead stakeholders, such as policymakers, to tailor communication and policy initiatives effectively.

Gathering robust data and conducting sound analyses, like this proportion comparison, can empower more informed decisions in the face of diverse and often conflicting interests surrounding offshore drilling projects.