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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

Browsing on the mobile device. A 2012 survey of 2,254 American adults indicates that \(17 \%\) of cell phone owners do their browsing on their phone rather than a computer or other device. (a) According to an online article, a report from a mobile research company indicates that 38 percent of Chinese mobile web users only access the internet through their cell phones. " Conduct a hypothesis test to determine if these data provide strong evidence that the proportion of Americans who only use their cell phones to access the internet is different than the Chinese proportion of \(38 \%\) (b) Interpret the p-value in this context. (c) Calculate a \(95 \%\) confidence interval for the proportion of Americans who access the internet on their cell phones, and interpret the interval in this context.

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.

Life rating in Greece. Greece has faced a severe economic crisis since the end of \(2009 .\) A Gallup poll surveyed 1,000 randomly sampled Greeks in 2011 and found that \(25 \%\) of them said they would rate their lives poorly enough to be considered "suffering". (a) Describe the population parameter of interest. What is the value of the point estimate of this parameter? (b) Check if the conditions required for constructing a confidence interval based on these data are met. (c) Construct a \(95 \%\) confidence interval for the proportion of Greeks who are "suffering". (d) Without doing any calculations, describe what would happen to the confidence interval if we decided to use a higher confidence level. (e) Without doing any calculations, describe what would happen to the confidence interval if we used a larger sample.

HIV in sub-Saharan Africa. In July 2008 the US National Institutes of Health announced that it was stopping a clinical study early because of unexpected results. The study population consisted of HIV-infected women in sub-Saharan Africa who had been given single dose Nevaripine (a treatment for HIV) while giving birth, to prevent transmission of HIV to the infant. The study was a randomized comparison of continued treatment of a woman (after successful childbirth) with Nevaripine vs. Lopinavir, a second drug used to treat HIV. 240 women participated in the study; 120 were randomized to each of the two treatments. Twenty-four weeks after starting the study treatment, each woman was tested to determine if the HIV infection was becoming worse (an outcome called virologic failure). Twenty-six of the 120 women treated with Nevaripine experienced virologic failure, while 10 of the 120 women treated with the other drug experienced virologic failure. (a) Create a two-way table presenting the results of this study. (b) State appropriate hypotheses to test for independence of treatment and virologic failure. (c) Complete the hypothesis test and state an appropriate conclusion. (Reminder: verify any necessary conditions for the test.)

Prop 19 in California. In a 2010 Survey USA poll, \(70 \%\) of the 119 respondents between the ages of 18 and 34 said they would vote in the 2010 general election for Prop \(19,\) which would change California law to legalize marijuana and allow it to be regulated and taxed. At a \(95 \%\) confidence level, this sample has an \(8 \%\) margin of error. Based on this information, determine if the following statements are true or false, and explain your reasoning." (a) We are \(95 \%\) confident that between \(62 \%\) and \(78 \%\) of the California voters in this sample support Prop \(19 .\) (b) We are \(95 \%\) confident that between \(62 \%\) and \(78 \%\) of all California voters between the ages of 18 and 34 support Prop \(19 .\) (c) If we considered many random samples of 119 California voters between the ages of 18 and 34 , and we calculated \(95 \%\) confidence intervals for each, \(95 \%\) of them will include the true population proportion of Californians who support Prop \(19 .\) (d) In order to decrease the margin of error to \(4 \%\), we would need to quadruple (multiply by 4) the sample size. (e) Based on this confidence interval, there is sufficient evidence to conclude that a majority of California voters between the ages of 18 and 34 support Prop \(19 .\)
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