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Chapter 23: Problem 10

The sample proportions of 9th- and 12th-graders who ate breakfast on all seven days before the survey are (a) \(\mathrm{p}^{\wedge \hat{p}_{9}}=0.396\) and \(\mathrm{p} \wedge \hat{p}{ }_{12}=0.338\). (b) \(\mathrm{p}^{\wedge \hat{p}} \hat{9}_{9}=0.368\) and \(\mathrm{p} \wedge \hat{p} \hat{12}_{12}=0.396\). (c) \(\mathrm{p}^{\wedge \hat{p}} \hat{9}_{9}=0.338\) and \(\mathrm{p}^{\wedge \hat{p}}{ }_{12}=0.368\).

Short Answer

Expert verified
The sample proportions for 9th- and 12th-graders are (a) \( \hat{p}_{9} = 0.396 \) and \( \hat{p}_{12} = 0.338 \).

Step by step solution

01

Identify provided sample proportions

We are given several pairs of sample proportions representing the likelihood that students ate breakfast on all seven days before a survey. The three options are:(a) \( \hat{p}_{9} = 0.396 \) and \( \hat{p}_{12} = 0.338 \),(b) \( \hat{p}_{9} = 0.368 \) and \( \hat{p}_{12} = 0.396 \),(c) \( \hat{p}_{9} = 0.338 \) and \( \hat{p}_{12} = 0.368 \).
02

Determine the correct sample proportions

The task is to refer directly to the exercise's requirement, identifying the correct correspondence between values and symbols as given without further computational steps. Considering the consistency of symbol uses across each pair, we aim to identify clearly the aligned statistical values.

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Key Concepts

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

Statistics Education
In the realm of statistics education, sample proportions are a fundamental topic that students encounter. This concept helps in understanding how often a particular event occurs within a sample as compared to the entire population. For instance, in the exercise provided, students are tasked with identifying correct sample proportions representing 9th- and 12th-graders eating breakfast every day over seven days.

Understanding sample proportions is crucial because they serve as estimates of true population proportions. In educational settings, students should focus on grasping how to compute and interpret these proportions accurately. Engaging with sample proportion tasks aids students in building a strong foundation for more complex statistical analyses.
  • Recognize that sample proportions are calculated as the number of successes (or occurrences) divided by the total sample size.
  • These values give a snapshot of larger population behaviors, facilitating predictions and interpretations.
  • Mastery of this concept enables students to analyze surveys, polls, and studies effectively.
Survey Analysis
Survey analysis involves the systematic collection and interpretation of data to understand general patterns or trends. In this specific context, the survey observed whether students consistently ate breakfast over a week. By examining sample proportions from the survey, you can derive insights about dietary habits among different age groups.

Sample proportions such as those given, \(\hat{p}_{9} = 0.396\)and\(\hat{p}_{12} = 0.338\),directly stem from survey data. Analysis of these proportions offers valuable information:
  • Identify behaviors or trends across demographic groups, such as different school grades.
  • Utilize this data to inform areas needing intervention or support.
  • Confirm or challenge existing assumptions about the population.
When analyzing surveys, it's essential to carefully evaluate the data source and collection method, ensuring reliable and valid findings.
Educational Statistics
Educational statistics is a method of employing statistical techniques to improve the understanding of educational systems and processes. By analyzing data like the breakfast consumption rates among different grades, educators and policy makers can develop strategies to enhance student well-being and performance.

Through statistical measurement, educational statistics provide insight into the efficacy of academic programs or health initiatives. For example, in the problem presented, if a significant disparity in breakfast eating habits between grades is noted, it might suggest the need for targeted nutritional programs.
  • Helps policymakers and educators to make evidence-based decisions.
  • Allows tracking of educational progress and the effectiveness of interventions.
  • Enables the identification of factors influencing student performance and health.
Applying educational statistics to practical scenarios ensures that educational practices are aligned with the actual needs and behaviors of the student population.

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Most popular questions from this chapter

An aerosolized vaccine for measles was developed in Mexico and has been used on more than 4 million children since 1980. Aerosolized vaccines have the advantages of being able to be administered by people without clinical training and do not cause injection-associated infections. Despite these advantages, data about efficacy of the aerosolized vaccines against measles compared to subcutaneously injection of the vaccine have been inconsistent. Because of this, a large randomized controlled study was conducted using children in India. The primary outcome was an immune response to measles measured 91 days after the treatments. Among the 785 children receiving the subcutaneous injection, 743 developed an immune response, while among the 775 children receiving the aerosolized vaccine, 662 developed an immune response. \({ }^{3}\) (a) Compute the proportion of subjects experiencing the primary outcome for both the aerosol and injection groups. (b) Can we safely use the large-sample confidence interval for comparing the proportion of children who developed an immune response to measles in the aerosol and injection groups? Explain. (c) Give a \(95 \%\) confidence interval for the difference between the proportion of children in the aerosol and injection groups who experienced the primary outcome. (d) The study described is an example of a noninferiority clinical trial intended to show that the effect of a new treatment, the aerosolized vaccine, is not worse than the standard treatment by more than a specified margin. 4 Specifically, is the percentage of children who developed an immune response for the aerosol treatment more than \(5 \%\) below the percentage for the subcutaneous injected vaccine? The five-percentage-point difference was based on previous studies and the fact that with a bigger difference the aerosolized vaccine would not provide the levels of protection necessary to achieve herd immunity. Using your answer in part (c), do you feel the investigators demonstrated the noninferiority of the aerosolized vaccine? Explain.

In the last 10 years, the prevalence of peanut allergies has doubled in Western countries. Is consumption or avoidance of peanuts in infants related to the development of peanut allergies in infants at risk? Subjects included infants between 4 and 11 months with severe eczema, egg allergy, or both, but who did not display a preexisting sensitivity to peanuts based on a skin-prick test. The infants were randomly assigned to either a treatment that avoided consuming peanut protein or a treatment in which at least 6 grams of peanut protein were consumed per week. The response was the presence or absence of peanut allergy at 60 months of age. In the avoidance group containing 263 infants, 36 had developed a peanut allergy at 60 months of age, while in the consumption group containing 266 infants, 5 had developed a peanut allergy. 13 (a) Despite the large sample sizes in both treatments, why should we not use the large-sample confidence interval for these data? (b) The plus four method adds one success and one failure in each sample. What are the sample sizes and counts of successes after you do this? (c) Give the plus four \(99 \%\) confidence interval for the difference in the probabilities of developing a peanut allergy for the avoidance and consumption treatments. What does your interval say about the comparison of these treatments in the context of the problem?

Bank employees from a large international bank were recruited with 67 assigned at random to a control group and the remaining 61 assigned to a treatment group. All subjects first completed a short online survey. After answering some general filler questions, the treatment group were asked seven questions about their professional background, such as "At which bank are you currently employed?" or "What is your function at this bank?" These are referred to as "identity priming" questions. The control group were asked seven innocuous questions unrelated to their profession, such as "How many hours a week on average do you watch television?" After the survey all subjects performed a coin-tossing task that required tossing any coin 10 times and reporting the results online. They were told they would win \(\$ 20\) for each head tossed for a maximum payoff of \(\$ 200\). Subjects were unobserved during the task, making it impossible to tell if a particular subject cheated. If the banking culture favors dishonest behavior, it was conjectured that it should be possible to trigger this behavior by reminding subjects of their profession. \({ }^{30}\) Here are the results. The first line gives the possible number of heads on 10 tosses, and the next two lines give the number of subjects that reported tossing this number of heads for the control and treatment groups, respectively (for example, 16 control subjects reported getting four heads). $$ \begin{array}{l|rrrrrrrrrrr} \hline \text { Number of heads } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\ \hline \text { Control group } & 0 & 0 & 1 & 8 & 16 & 17 & 14 & 6 & 2 & 1 & 2 \\\ \hline \text { Treatment group } & 0 & 0 & 2 & 4 & 8 & 14 & 15 & 7 & 6 & 0 & 5 \\\ \hline \end{array} $$ If a subject is cheating, we would expect them to report doing better than chance, or tossing six or more heads. (a) Find the proportion of subjects in each group that reported tossing six or more heads. (b) Test the hypotheses that the proportions reporting tossing six or more heads in the two groups are the same against the appropriate alternative. Explain your conclusions in the context of the problem, being sure to relate this to the researcher's conjecture.

In recent years, a number of new commercial online services have emerged that have altered some aspect of people's lives. Of these, ride-hailing apps provide a good example of this new on-demand economy. A Pew Internet survey in 2016 examined several demographic variables of users of ride-hailing apps such as Uber or Lyft, including education, age, race, and income. Of the 4787 adults included in the survey, 2369 were college graduates, of which 687 had used a ride-hailing app. Among the 2418 adults who had not completed college, 268 had used a ride-hailing app. \({ }^{23}\) Is there good evidence that the proportion of adults who have used a ride-hailing app is different between college graduates and those without college degrees?

Many teens have posted profiles on a socialnetworking website. A sample survey in 2007 asked a random sample of teens with online profiles if they included false information in their profiles. Of 170 younger teens (aged 12 to 14\(), 117\) said yes. Of 317 older teens (aged 15 to 17), 152 said yes. 16 (a) Do these samples satisfy the guidelines for the large-sample confidence interval? (b) Give a 95\% confidence interval for the difference between the proportions of younger and older teens who include false information in their online profiles.
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