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Surveying Students. You are planning a survey of students at a large university to determine what proportion favor an increase in student fees to support an expansion of the student newspaper. Using records provided by the registrar, you can select a random sample of students. You will ask each student in the sample whether he or she is in favor of the proposed increase. Your budget will allow a sample of 100 students. a. For a sample of size 100, construct a table of the margins of error for \(95 \%\) confidence intervals when \(\widehat{p}\) takes the values \(0.1,0.3,0.5,0.7\), and \(0.9\). b. A former editor of the student newspaper offers to provide funds for a sample of size 500 . Repeat the margin of error calculations in part (a) for the larger sample size. Then write a short thank-you note to the former editor, describing how the larger sample size will improve the results of the survey. In responding to \(\underline{\text { Exercises } 22.35}\) through \(\underline{22.41}\), follow the Plan, Solve, and Conclude steps of the four- step process.

Step by step solution

01

Understand the Problem

We're asked to find the margin of error for a 95% confidence interval for different sample sizes and proportions. The formula for margin of error (ME) is \( ME = z \times \sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}} \), where \( z \) is the z-score (1.96 for 95% confidence), \( \widehat{p} \) is the sample proportion, and \( n \) is the sample size.

02

Calculate the Margin of Error for Sample Size 100

We calculate the margins of error for \( n = 100 \) and \( \widehat{p} = \{0.1, 0.3, 0.5, 0.7, 0.9\} \). For example, for \( \widehat{p} = 0.1 \), the margin of error is \( ME = 1.96 \times \sqrt{\frac{0.1 \times 0.9}{100}} \approx 0.0588 \). Similarly, calculate for other proportions.

03

Present the Table for Sample Size 100

Proportion (\(\widehat{p}\)) | Margin of Error (\(n = 100\))---0.1 | 0.05880.3 | 0.08920.5 | 0.09800.7 | 0.08920.9 | 0.0588

04

Calculate the Margin of Error for Sample Size 500

Now, calculate the margins of error for \( n = 500 \) using the same formula. For \( \widehat{p} = 0.1 \), the margin of error is \( ME = 1.96 \times \sqrt{\frac{0.1 \times 0.9}{500}} \approx 0.0262 \). Repeat for other proportions.

05

Present the Table for Sample Size 500

Proportion (\(\widehat{p}\)) | Margin of Error (\(n = 500\))---0.1 | 0.02620.3 | 0.03970.5 | 0.04390.7 | 0.03970.9 | 0.0262

06

Write a Thank-You Note

Dear Former Editor, Thank you for your generous contribution, which allows us to increase our sample size to 500 students. This increase significantly reduces the margin of error in our survey results, providing more precise estimates of student opinions on the student fee increase. Sincerely, [Your Name]

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Key Concepts

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

Margin of Error

When conducting a survey, the margin of error (ME) is crucial as it tells us the range where our true population parameter is likely to fall. It accounts for the sampling variability that comes from asking only a part of the population. The formula to compute it is:\[ ME = z \times \sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}} \]where:

• \( z \) is the z-score related to your desired confidence level (for example, 1.96 for 95% confidence)
• \( \widehat{p} \) is the sample proportion
• \( n \) is the sample size

The margin of error decreases as the sample size increases because larger samples tend to be more representative of the population. Understanding ME helps in interpreting survey results and assessing their reliability.

Confidence Interval

The confidence interval (CI) gives us a range of values that we believe includes the true population parameter with a certain degree of certainty. A 95% CI suggests that if you were to take 100 different samples, about 95 of them would contain the true population proportion.To calculate a confidence interval, you typically use:
\[ \widehat{p} \pm ME \]where \( \widehat{p} \) is the sample proportion and \( ME \) is the margin of error. Confidence intervals provide a way to understand the accuracy of your sample result. They depict how much uncertainty there is in the estimate of the population parameter, offering a balance between precision and reliability.

Sample Size

Sample size, represented as \( n \), plays a significant role in survey accuracy. Larger sample sizes generally yield more accurate estimates by reducing the margin of error. In the exercise, a sample size of 100 was initially considered, which provided specific margins of error for various proportions. By increasing the sample size to 500, the precision improved noticeably, reflected by smaller margins of error.A good approach when planning your survey is to work within your budget while striving for the greatest possible sample size, as this optimizes the reliability of your findings without unnecessary expense.

Proportion

Proportion, often represented by \( \widehat{p} \), is the fraction of the sample that expresses a particular characteristic. For instance, in the exercise, it represents the proportion of students in favor of the fee increase.Sample proportion is a fundamental statistic when determining the margin of error and confidence interval. It influences the width of the confidence interval related to the survey's findings. As sample size increases, the estimate of this proportion becomes more reliable, with less variability, making survey results potentially more indicative of true population values.

Random Sample

Random sampling is a method where each member of a population has an equal chance of being included in the sample. This technique is crucial in obtaining a representative sample that can provide unbiased statistics about the whole population. For the student survey, using a random sample ensures that different student demographics are fairly represented, reducing the risk of bias. Employing a random sampling technique enhances the external validity of the survey results, meaning the findings are more likely to be generalizable to the entire student body, which is what the survey ultimately aims to achieve.