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Chapter 11: Problem 16

The article referenced in the previous exercise also reported that \(53 \%\) of the Republicans surveyed indicated that they were opposed to making women register for the draft. Would you use the large-sample test for a difference in population proportions to test the hypothesis that a majority of Republicans are opposed to making women register for the draft? Explain why or why not.

Short Answer

Expert verified
No, the large-sample test for a difference in population proportions would not be appropriate to test the hypothesis that a majority of Republicans are opposed to making women register for the draft, as we have only one proportion given (53% of Republicans opposed) and cannot compare two populations. Instead, we should use the large-sample test for a single proportion with the null hypothesis (H0) being that the proportion is equal to 0.5 and the alternative hypothesis (H1) being that the proportion is greater than 0.5. However, we need the sample size from the original exercise to proceed with the test.

Step by step solution

01

State the Hypothesis

H0: p = 0.5 H1: p > 0.5
02

Calculate the Test Statistic

To calculate the test statistic, use the following formula: \[z = \frac{(\hat{p} - p_0)}{\sqrt{\frac{p_0q_0}{n}}}\] where \(\hat{p}\) is the sample proportion (53%), \(p_0\) is the null hypothesis proportion (0.5), \(q_0 = 1 - p_0\), and \(n\) is the sample size. We need the sample size (n) from the original exercise to proceed with the test.
03

Find the P-value

Once the test statistic (z) is calculated, find the P-value by looking up the area to the right of the z-value in a standard normal distribution table (since the alternative hypothesis is a "greater than" hypothesis).
04

Draw Conclusion

Compare the P-value to the significance level (\(\alpha\), usually 0.05) to determine if the null hypothesis can be rejected. - If P-value 鈮� 伪, reject the null hypothesis and conclude that there is enough evidence to support the claim that a majority of Republicans are opposed to making women register for the draft. - If P-value > 伪, fail to reject the null hypothesis and conclude that there is not enough evidence to support the claim that a majority of Republicans are opposed to making women register for the draft. Please note that the exercise can only be fully completed with the provided sample size from the original exercise.

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Key Concepts

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

Population Proportions
In hypothesis testing, population proportions refer to the fraction or percentage of members in a population that holds a particular attribute. In the context of the exercise, a population proportion might represent the percentage of Republicans opposed to making women register for the draft.
Here are some key points to consider:
  • Proportions are usually denoted by the symbol \( p \), indicating the true proportion in the overall population, while \( \hat{p} \) represents the sample proportion.
  • In this scenario, the sample proportion (\( \hat{p} = 0.53 \)) signifies that 53% of Republicans in the sample oppose the draft for women.
  • To draw conclusions about the population proportion, researchers use statistical tests to determine if the observed sample proportion (\( \hat{p} \)) is significantly different from a hypothesized population proportion (\( p_0 \)).
Understanding these concepts helps in analyzing whether a majority stance exists within a specified group. By comparing \( \hat{p} \) with \( p_0 \), we can statistically infer if the perceived beliefs of the population align with the hypothesized stance.
Significance Level
The significance level, often represented by the Greek letter \( \alpha \), plays a crucial role in hypothesis testing. It sets a threshold for determining whether your test results are statistically significant. Typically, a significance level of 0.05 is used in many disciplines.
  • A significance level of 0.05 implies a 5% risk of concluding that a difference exists when there is no actual difference (type I error).
  • By choosing \( \alpha = 0.05 \), you accept a 5% chance of mistakenly rejecting the null hypothesis.
  • The smaller the \( \alpha \), the stricter the test's criteria, making it less likely that findings are due to chance.
In the exercise, the significance level is utilized to compare the P-value derived from the hypothesis test. If this P-value is less than or equal to \( \alpha \), it indicates there is enough evidence to conclude that a majority indeed has the stated belief.
Null and Alternative Hypotheses
In any hypothesis testing scenario, formulating your null and alternative hypotheses is essential.
  • The null hypothesis \( H_0 \) is a statement of no effect or no difference, asserting that the proportion is not greater than 50% (\( p = 0.5 \)).
  • For the alternative hypothesis \( H_1 \), it assumes that there is a change or a difference, suggesting that more than 50% of Republicans oppose registering women for the draft (\( p > 0.5 \)).
  • These hypotheses serve as foundations for testing and guide the type of statistical tests applied.
The main idea behind these hypotheses is to verify whether the sample's observations can provide sufficient evidence to reject the null hypothesis, favoring the alternative. Once established, a statistical test can begin to determine if the data supports the null or the alternative hypothesis.
Sample Size
Sample size is a significant factor in statistical testing, influencing the reliability of your inferential statistics. It refers to the number of observations or individual data points collected in the study.
  • A larger sample size generally provides more accurate estimations of the population proportion because it reduces sampling error.
  • The formula for the test statistic in population proportion testing includes \( n \), representing the sample size, affecting the variability of the sample proportion \( \hat{p} \).
  • There should be a sufficient sample size to ensure the validity of normal approximations used in calculating test results, often requiring both \( np \) and \( n(1-p) \) to be greater than 5.
In the exercise, the sample size is crucial to performing reliable hypothesis testing for population proportions. A properly sized sample ensures that results from statistical tests are both valid and reliable.

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Most popular questions from this chapter

The article "More Teen Drivers See Marijuana as OK; It's a Dangerous Trend" (USA TODAY, February 23,2012 ) describes two surveys of U.S. high school students. One survey was conducted in 2009 and the other was conducted in \(2011 .\) In \(2009,78 \%\) of the people in a representative sample of 2300 students said marijuana use is very distracting or extremely distracting to their driving. In \(2011,70 \%\) of the people in a representative sample of 2294 students answered this way. Use the five-step process for estimation problems \(\left(\mathrm{EMC}^{3}\right)\) to construct and interpret a \(99 \%\) large-sample confidence interval for the difference in the proportion of high school students who believed marijuana was very distracting or extremely distracting in 2009 and this proportion in 2011 .

A report in USA TODAY described an experiment to explore the accuracy of wearable devices designed to measure heart rate ("Wearable health monitors not always reliable, study shows," USA TODAY, October 12,2016\()\). The researchers found that when 50 volunteers wore an Apple Watch to track heart rate as they walked, jogged, and ran quickly on a treadmill for three minutes, the results were accurate compared with an EKG 92\% of the time. When 50 volunteers wore a Fitbit Charge, the heart rate results were accurate \(84 \%\) of the time. a. Explain why the data from this study should not be analyzed using a large- sample hypothesis test for a difference in two population proportions. b. Carry out a hypothesis test to determine if there is convincing evidence that the proportion of accurate results for people wearing an Apple Watch is greater than this proportion for those wearing a Fitbit Charge. Use the Shiny app "Randomization Test for Two Proportions" to report an approximate \(P\) -value and use it to reach a decision in the hypothesis test. Remember to interpret the results of the test in context. c. Use the Shiny app "Bootstrap Confidence Interval for Difference in Two Proportions" to obtain a \(95 \%\) bootstrap confidence interval for the difference in the population proportions of accurate results for people wearing an Apple Watch and those wearing a Fitbit Charge. Interpret the interval in the context of the research.

The report titled "Digital Democracy Survey" (2016, www .deloitte.com/us/tmttrends, retrieved December 16,2016 ) stated that \(31 \%\) of the people in a representative sample of adult Americans age 33 to 49 rated a landline telephone among the three most important services that they purchase for their home. In a representative sample of adult Americans age 50 to \(68,48 \%\) rated a landline telephone as one of the top three services they purchase for their home. Suppose that the samples were independently selected and that the sample size was 600 for the 33 to 49 age group sample and 650 for the 50 to 68 age group sample. Does this data provide convincing evidence that the proportion of adult Americans age 33 to 49 who rate a landline phone in the top three is less than this proportion for adult Americans age 50 to 68 ? Test the relevant hypotheses using \(\alpha=0.05\)

In a survey of mobile phone owners, \(53 \%\) of iPhone users and \(42 \%\) of Android phone users indicated that they upgraded their phones at least every two years ("Americans Split on How Often They Upgrade Their Smartphones," July 8, \(2015,\) www.gallup.com, retrieved December 15,2016 ). The reported percentages were based on large samples that were thought to be representative of the population of iPhone users and the population of Android phone users. The sample sizes were 8234 for the iPhone sample and 6072 for the Android phone sample. Suppose you want to decide if there is evidence that the proportion of iPhone owners who upgrade their phones at least every two years is greater than this proportion for Android phone users. (Hint: See Example 11.5.) a. What hypotheses should be tested to answer the question of interest? b. Are the two samples large enough for the large-sample test for a difference in population proportions to be appropriate? Explain. c. Based on the following Minitab output, what is the value of the test statistic and what is the value of the associated \(P\) -value? If a significance level of 0.01 is selected for the test, will you reject or fail to reject the null hypothesis? d. Interpret the result of the hypothesis test in the context of this problem.

Women diagnosed with breast cancer whose tumors have not spread may be faced with a decision between two surgical treatments -mastectomy (removal of the breast) or lumpectomy (only the tumor is removed). In a long-term study of the effectiveness of these two treatments, 701 women with breast cancer were randomly assigned to one of two treatment groups. One group received mastectomies and the other group received lumpectomies and radiation. Both groups were followed for 20 years after surgery. It was reported that there was no statistically significant difference in the proportion surviving for 20 years for the two treatments (Associated Press, October \(17,\) 2002). What hypotheses do you think the researchers tested in order to reach the given conclusion? Did the researchers reject or fail to reject the null hypothesis?
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