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Chapter 3: Problem 31

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.

Short Answer

Expert verified
College grads: ~35.16% support; Non-college grads: ~33.93% support. Hypothesis test: No strong evidence of a significant difference.

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 \(卤1.96\) 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.

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Most popular questions from this chapter

Gender and color preference. A 2001 study asked 1,924 male and 3,666 female undergraduate college students their favorite color. A \(95 \%\) confidence interval for the difference between the proportions of males and females whose favorite color is black ( \(p_{\text {male }}-p\) female \()\) was calculated to be (0.02,0.06) . Based on this information, determine if the following statements are true or false, and explain your reasoning for each statement you identify as false. 2 (a) We are \(95 \%\) confident that the true proportion of males whose favorite color is black is \(2 \%\) lower to \(6 \%\) higher than the true proportion of females whose favorite color is black. (b) We are \(95 \%\) confident that the true proportion of males whose favorite color is black is \(2 \%\) to \(6 \%\) higher than the true proportion of females whose favorite color is black. (c) \(95 \%\) of random samples will produce \(95 \%\) confidence intervals that include the true difference between the population proportions of males and females whose favorite color is black. (d) We can conclude that there is a significant difference between the proportions of males and females whose favorite color is black and that the difference between the two sample proportions is too large to plausibly be due to chance. (e) The \(95 \%\) confidence interval for \(\left(p_{\text {female }}-p_{\text {male }}\right)\) cannot be calculated with only the information given in this exercise.

Sleep deprived transportation workers. The National Sleep Foundation conducted a survey on the sleep habits of randomly sampled transportation workers and a control sample of non-transportation workers. The results of the survey are shown below. \begin{tabular}{lccccc} & \multicolumn{4}{c} { Transportation Professionals } \\ \cline { 3 - 6 } & & & Truck & Train & Bux/1axi/Limo \\ & Control & Pilots & Drivers & Operators & Drivers \\ \hline Less than 6 hours of sleep & 35 & 19 & 35 & 29 & 21 \\ 6 to 8 hours of sleep & 193 & 132 & 117 & 119 & 131 \\ More than 8 hours & 64 & 51 & 51 & 32 & 58 \\ \hline Total & 292 & 202 & 203 & 180 & 210 \end{tabular} Conduct a hypothesis test to evaluate if these data provide evidence of a difference between the proportions of truck drivers and non-transportation workers (the control group) who get less than 6 hours of sleep per day, i.e. are considered sleep deprived.

3.29 Offshore drilling, Part 1. A 2010 survey asked 827 randomly sampled registered voters in California "Do you support? Or do you oppose? Drilling for oil and natural gas off the Coast of California? Or do you not know enough to say?" Below is the distribution of responses, separated based on whether or not the respondent graduated from college. (a) What percent of college graduates and what percent of the non-college graduates in this sample do not know enough to have an opinion on drilling for oil and natural gas off the Coast of California? \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} the (b) Conduct a hypothesis test to determine if data provide strong evidence that the proportion of college graduates who do not have an opinion on this issue is different than that of non-college graduates.

Open source textbook. A professor using an open source introductory statistics book predicts that \(60 \%\) of the students will purchase a hard copy of the book, \(25 \%\) will print it out from the web, and \(15 \%\) will read it online. At the end of the semester he asks his students to complete a survey where they indicate what format of the book they used. Of the 126 students, 71 said they bought a hard copy of the book, 30 said they printed it out from the web, and 25 said they read it online. (a) State the hypotheses for testing if the professor's predictions were inaccurate. (b) How many students did the professor expect to buy the book, print the book, and read the book exclusively online? (c) This is an appropriate setting for a chi-square test. List the conditions required for a test and verify they are satisfied. (d) Calculate the chi-squared statistic, the degrees of freedom associated with it, and the p-value. (e) Based on the p-value calculated in part (d), what is the conclusion of the hypothesis test? Interpret your conclusion in this context.

True or false, Part. II. Determine if the statements below are true or false. For each false statement, suggest an alternative wording to make it a true statement. (a) As the degrees of freedom increases, the mean of the chi-square distribution increases. (b) If you found \(X^{2}=10\) with \(d f=5\) you would fail to reject \(H_{0}\) at the \(5 \%\) significance level. (c) When finding the p-value of a chi-square test, we always shade the tail areas in both tails. (d) As the degrees of freedom increases, the variability of the chi-square distribution decreases.
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