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Chapter 6: Problem 43

When we want \(95 \%\) confidence and use the conservative estimate of \(p=0.5,\) we can use the simple formula \(n=1 /(M E)^{2}\) to estimate the sample size needed for a given margin of error ME. In Exercises 6.40 to 6.43, use this formula to determine the sample size needed for the given margin of error. A margin of error of 0.05 .

Short Answer

Expert verified
The required sample size to achieve a margin of error of 0.05 with the conservative estimate of \(p=0.5\) and \(95\%\) confidence is \(400\).

Step by step solution

01

Understand the Given Values

The task mentions a target confidence level of \(95 \% \) and a conservative estimate for proportion \(p=0.5\). We are given a formula to estimate the sample size needed for a given margin of error \(ME\), i.e., \(n=1 /(M E)^{2}\). The given margin of error is \(0.05\).
02

Substitute values into formula

Substitute \(ME = 0.05\) into the formula \(n=1 /(M E)^{2}\). This gives \(n = 1 /(0.05)^{2}\).
03

Solve for n

Calculate the value for \(n\). Using the formula from step 2 gives \(n = 1 /(0.05)^{2} = 400\). Thus, a sample size of 400 is needed to achieve a margin of error of 0.05.

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Key Concepts

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

Confidence Interval
In statistics, a confidence interval is a range of values that's used to estimate an unknown population parameter. Typically, this parameter could be the population mean or proportion. The confidence interval gives us a span that likely contains the true value of this parameter with a certain level of confidence, like 95% in many cases.
For example, if we calculate a 95% confidence interval for a population mean, we're saying that there's a 95% chance that our interval contains the true mean. The higher the confidence level, the wider the interval will be, as we need to be more certain that the true value lies within our estimated range.
  • Interpretation of a Confidence Interval: It is crucial to remember that the confidence interval is for the unknown parameter, not the likely values of a new observation.
  • Role in Sample Size Calculation: The desired confidence level is directly related to determining how large a sample size should be for accurate estimation.
Confidence intervals are powerful tools in statistics, providing insights into the reliability of our estimates and helping in making informed decisions based on data.
Margin of Error
The margin of error is a statistic expressing the amount of random sampling error in a survey's results. It represents variability, or uncertainty, in the estimate.
The margin of error gives the radius (half the width) of the confidence interval for an unknown population parameter. For a given sample size, the margin of error is the maximum expected difference between the true population parameter and a sample estimate of that parameter.
  • Example: If the margin of error is 0.05 for a sample proportion, it means we are 95% confident that the true proportion lies within 0.05 of the observed proportion.
  • Importance: The margin of error is crucial in communicating the expected accuracy of an estimate to stakeholders, such as researchers or the public.
Understanding the margin of error is key to interpreting statistical results and helps ensure the reliability and validity of conclusions drawn from data.
Statistical Estimation
Statistical estimation involves using sample data to estimate population parameters. Two primary methods of estimation include point estimation and interval estimation. Point estimation provides a single value as an estimate of the parameter. In contrast, interval estimation, like a confidence interval, gives a range of plausible values.
Statistical estimation is foundational in making predictions and decisions based on data, and it leverages sample data to generalize findings to a larger population. This process inherently involves some degree of uncertainty, since we infer characteristics of a whole population from a sample.
  • Basic Types of Estimation:
    • Point Estimation: Provides the most likely value of a parameter.
    • Interval Estimation: Offers a range within which the parameter likely lies, as seen in confidence intervals.
  • Practical Use: Estimation helps in various domains, from scientific research to business analytics and beyond.
Estimation allows us to make informed predictions and decisions based on limited data, balancing precision with practicality.

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Most popular questions from this chapter

Metal Tags on Penguins and Survival Data 1.3 on page 10 discusses a study designed to test whether applying metal tags is detrimental to penguins. One variable examined is the survival rate 10 years after tagging. The scientists observed that 10 of the 50 metal tagged penguins survived, compared to 18 of the 50 electronic tagged penguins. Construct a \(90 \%\) confidence interval for the difference in proportion surviving between the metal and electronic tagged penguins \(\left(p_{M}-p_{E}\right)\). Interpret the result.

Use a t-distribution to find a confidence interval for the difference in means \(\mu_{1}-\mu_{2}\) using the relevant sample results from paired data. Give the best estimate for \(\mu_{1}-\) \(\mu_{2},\) the margin of error, and the confidence interval. Assume the results come from random samples from populations that are approximately normally distributed, and that differences are computed using \(d=x_{1}-x_{2}\). A \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) using the paired difference sample results \(\bar{x}_{d}=3.7, s_{d}=\) 2.1, \(n_{d}=30\)

In Exercises 6.103 and 6.104 , find a \(95 \%\) confidence interval for the mean two ways: using StatKey or other technology and percentiles from a bootstrap distribution, and using the t-distribution and the formula for standard error. Compare the results. Mean distance of a commute for a worker in Atlanta, using data in Commute Atlanta with \(\bar{x}=\) 18.156 miles, \(s=13.798,\) and \(n=500\)

In Exercises 6.34 to 6.36, we examine the effect of different inputs on determining the sample size needed to obtain a specific margin of error when finding a confidence interval for a proportion. Find the sample size needed to give, with \(95 \%\) confidence, a margin of error within \(\pm 6 \%\) when estimating a proportion. Within \(\pm 4 \%\). Within \(\pm 1 \%\). (Assume no prior knowledge about the population proportion \(p\).) Comment on the relationship between the sample size and the desired margin of error.

Statistical Inference in Babies Is statistical inference intuitive to babies? In other words, are babies able to generalize from sample to population? In this study, \(1 \quad 8\) -month-old infants watched someone draw a sample of five balls from an opaque box. Each sample consisted of four balls of one color (red or white) and one ball of the other color. After observing the sample, the side of the box was lifted so the infants could see all of the balls inside (the population). Some boxes had an "expected" population, with balls in the same color proportions as the sample, while other boxes had an "unexpected" population, with balls in the opposite color proportion from the sample. Babies looked at the unexpected populations for an average of 9.9 seconds \((\mathrm{sd}=4.5\) seconds) and the expected populations for an average of 7.5 seconds \((\mathrm{sd}=4.2\) seconds). The sample size in each group was \(20,\) and you may assume the data in each group are reasonably normally distributed. Is this convincing evidence that babies look longer at the unexpected population, suggesting that they make inferences about the population from the sample? (a) State the null and alternative hypotheses. (b) Calculate the relevant sample statistic. (c) Calculate the t-statistic.
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