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Exercises 71 to 74 refer to the following setting. Coaching companies claim that their courses can raise the SAT scores of high school students. Of course, students who retake the SAT without paying for coaching generally raise their scores. A random sample of students who took the SAT twice found 427 who were coached and 2733 who were uncoached.\(^{44}\) Starting with their Verbal scores on the first and second tries, we have these summary statistics: Coaching and SAT scores (10.1) What proportion of students who take the SAT twice are coached? To answer this question, Jannie decides to construct a 99% confidence interval. Her work is shown below. Explain what鈥檚 wrong with Jannie鈥檚 method. $$hat{p}_{1}=\frac{427}{3160}=0.135=\underset{\text { who were coached }}{\text { proportion of students }}$$ $$\hat{p}_{2}=\frac{2733}{3160}=0.865=\underset{\text { who weren't coached }}{\text { who weren't coached }}$$ $$\mathrm{A} 99 \% \mathrm{Cl} \text { for } p_{1}-p_{2} \mathrm{is}$$ $$\begin{aligned}(0.135-0.865) \pm 2.575 \sqrt{\frac{0.135(0.865)}{3160}} &+\frac{0.865(0.135)}{2733} \\ &=-0.73 \pm 0.022=(-0.752,-0.708) \end{aligned}$$ We are 99% confident that the proportion of students taking the SAT twice who are coached is between 71 and 75 percentage points lower than students who aren鈥檛 coached.

Step by step solution

01

Understanding the problem

We need to determine what proportion of the students who retake the SAT are coached. Jannie is trying to use a confidence interval to compare the proportion of coached students vs. uncoached students.

02

Review of Jannie's data

Jannie calculates the proportion of coached students as \( \hat{p}_1 = \frac{427}{3160} = 0.135 \). This means approximately 13.5% of students who took the SAT twice were coached.

03

Analyzing Jannie's approach

Jannie is calculating a confidence interval for the difference in proportions, \( p_1 - p_2 \), to see how much lower the proportion of coached students is compared to uncoached students, which isn鈥檛 the intention of the problem.

04

Correct Approach to Calculate Proportion

The correct task is to calculate the proportion of students who were coached, which Jannie computed correctly as 0.135.

05

Identifying Mistakes

Jannie incorrectly used a formula for the difference of proportions to construct a confidence interval, which is not necessary or correct for simply finding the proportion of coached students.

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Key Concepts

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

SAT Coaching Impact

Coaching programs which aim to boost SAT scores claim that students can make significant improvements. The SAT, a standardized test commonly used for college admissions in the United States, often sees students taking it multiple times, sometimes opting for coaching services to improve their scores. Coaching can:

• Provide tailored study plans
• Offer test-taking strategies
• Help reduce anxiety through practice exams

Understanding the impact of such coaching requires comparing the performance of those who are coached versus those who are not. In the exercise we are discussing, we see that 427 out of a total of 3160 students were coached. This generates intriguing questions about the actual benefits and effectiveness of paid coaching for standardized tests.

Proportion Calculation

Proportion calculation is a fundamental concept in statistics used to determine the fraction of a total that possesses a particular attribute. In our exercise, we focus on the proportion of students who are coached for their SAT retakes.The calculation is straightforward: \[\hat{p} = \frac{\text{Number of Coached Students}}{\text{Total Students}} = \frac{427}{3160} = 0.135\]This calculation shows that approximately 13.5% of students were coached for their SAT retakes. Proportion calculation helps us to break down data into understandable and more interpretable parts. It forms a base for further analysis like constructing confidence intervals or performing error analysis.

Error Analysis

Error analysis is crucial in statistics to ensure the accuracy and reliability of results derived from data. In the exercise, an error was made in using the wrong formula to calculate a confidence interval. Jannie incorrectly attempted to calculate a confidence interval for the difference in proportions between the coached and uncoached groups.
To understand errors better:

• Determine the goal - calculating a single proportion versus comparing proportions.
• Use the correct formulas specific to the calculations needed.
• Review results to check whether they logically align with expectated outcomes.

Jannie's mistake stemmed from using a comparison formula when she should have only calculated a straightforward proportion, which is a reminder to always align methods with objectives.

Statistics Education

Statistics is a powerful field that allows us to make inferences from data and apply them to real-world situations. In education, statistics helps students understand how to analyze data and draw appropriate conclusions. Jannie's exercise serves as a learning opportunity by displaying what can go wrong and the importance of understanding statistical concepts deeply.
Key teachings from statistics education include:

• Understanding basic calculations like proportions and means.
• Applying the right statistical techniques for varying problems.
• Interpreting results within the correct context.

By delving into these aspects, students learn to appreciate the nuances of statistical thinking, leading to smarter analysis and decision-making in many academic and professional areas.