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Chapter 7: Problem 4

You want to conduct a survey to determine the proportion of people who favor a proposed tax policy. How does increasing the sample size affect the size of the margin of error?

Short Answer

Expert verified
Increasing sample size decreases the margin of error, making estimates more precise.

Step by step solution

01

Understanding the Margin of Error

The margin of error in a survey is the range within which the true population parameter is expected to fall. It is calculated using the formula \( ME = z \times \sqrt{ \frac{p(1-p)}{n} } \), where \(z\) is the z-score corresponding to the desired confidence level, \(p\) is the sample proportion, and \(n\) is the sample size.
02

Role of Sample Size in Margin of Error

The formula shows that the margin of error is inversely proportional to the square root of the sample size \(n\). This means that as the sample size increases, the denominator of the fraction inside the square root becomes larger, which results in a smaller overall margin of error.
03

Conclusion: Impact of Increasing Sample Size

Increasing the sample size decreases the margin of error, making your survey estimate more precise. A larger sample size provides more reliable data about the population parameter by reducing the variability in the estimate.

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Key Concepts

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

Margin of Error
The margin of error quantifies the uncertainty in a survey's results. It tells you how much the survey results might differ from the true population value. This is crucial because no survey can perfectly predict the population parameter due to sampling variability.
The margin of error is calculated using the formula: \[ ME = z \times \sqrt{ \frac{p(1-p)}{n} } \]
  • \( z \): z-score corresponds to your desired confidence level.
  • \( p \): sample proportion, the fraction of respondents favoring the policy.
  • \( n \): sample size, or the number of people surveyed.
Understanding the margin of error helps you interpret how reliable and close your sample results are to the actual population values.
Sample Size Effect
The sample size is a critical factor in determining the reliability of your survey results. In simple terms, the sample size is the number of individuals or observations included in your survey.
The effect of sample size on margin of error is best described as inversely proportional. This means:
  • As the sample size increases, the margin of error decreases.
  • A larger sample size gives you a more accurate estimate of the population parameter.
Consider a small survey sample - it might capture extreme views purely by chance. But, with a large enough sample, those extremes level out, and you get a clearer image of the population, minimizing error.
Confidence Level
The confidence level is a measure of how certain you are that the true population parameter falls within your survey's margin of error. It is often expressed as a percentage, like 95% or 99%.
If you have a 95% confidence level, it means you expect the true proportion to be in your margin of error range 95 times out of 100 surveys conducted.
The choice of confidence level directly influences the z-score in the margin of error formula:
  • Higher confidence levels require a higher z-score.
  • This increases the margin of error but reflects greater certainty.
Balancing confidence level and margin of error is crucial in survey design, ensuring reliable yet practical results.
Population Proportion
Population proportion represents the fraction of the total population that holds a certain attribute or opinion, such as supporting a policy.
In surveys, this is estimated using the sample proportion, derived from those surveyed. In the margin of error formula, the term \( p(1-p) \) represents this variability in proportion:
  • \( p \) is the estimated proportion from your sample.
  • \( 1-p \) indicates those not supporting the policy.
It shows the likelihood that any given sample may have overrepresented or underrepresented the true population opinion.
Understanding population proportion helps predict how representative and accurate your survey findings are compared to the entire population.
Statistical Accuracy
Statistical accuracy refers to how close your survey results are to the actual population parameters. Accuracy depends heavily on factors like the margin of error, sample size, and confidence level.
To improve statistical accuracy in your surveys:
  • Increase sample size, which reduces the margin of error.
  • Choose an appropriate confidence level.
  • Ensure your sample is representative of the entire population.
High accuracy means lower chances of errors or bias and more reliable information to support decisions based on the survey data. It鈥檚 a clear measure of confidence in the survey's conclusions.

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Most popular questions from this chapter

Baseball: Home Run Percentage The home run percentage is the number of home runs per 100 times at bat. A random sample of 43 professional baseball players gave the following data for home run percentages (Reference: The Baseball Encyclopedia, Macmillan). $$ \begin{array}{llllllllll} 1.6 & 2.4 & 1.2 & 6.6 & 2.3 & 0.0 & 1.8 & 2.5 & 6.5 & 1.8 \\ 2.7 & 2.0 & 1.9 & 1.3 & 2.7 & 1.7 & 1.3 & 2.1 & 2.8 & 1.4 \\ 3.8 & 2.1 & 3.4 & 1.3 & 1.5 & 2.9 & 2.6 & 0.0 & 4.1 & 2.9 \\ 1.9 & 2.4 & 0.0 & 1.8 & 3.1 & 3.8 & 3.2 & 1.6 & 4.2 & 0.0 \\ 1.2 & 1.8 & 2.4 & & & & & & & \end{array} $$ (a) Use a calculator with mean and standard deviation keys to verify that \(\bar{x} \approx 2.29\) and \(s \approx 1.40\). (b) Compute a \(90 \%\) confidence interval for the population mean \(\mu\) of home run percentages for all professional baseball players. Hint: If you use Table 6 of Appendix II, be sure to use the closest \(d . f\). that is smaller. (c) Compute a \(99 \%\) confidence interval for the population mean \(\mu\) of home run percentages for all professional baseball players. (d) Interpretation The home run percentages for three professional players are Tim Huelett, \(2.5 \quad\) Herb Hunter, \(2.0 \quad\) Jackie Jensen, \(3.8\) Examine your confidence intervals and describe how the home run percentages for these players compare to the population average. (e) Check Requirements In previous problems, we assumed the \(x\) distribution was normal or approximately normal. Do we need to make such an assumption in this problem? Why or why not? Hint: See the central limit theorem in Section \(6.5\).

Expand Your Knowledge: Alternate Method for Confidence Intervals When \(\sigma\) is unknown and the sample is of size \(n \geq 30\), there are two methods for computing confidence intervals for \(\mu\). Method 1: Use the Student's \(t\) distribution with d.f. \(=n-1\). This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method. Method 2: When \(n \geq 30\), use the sample standard deviation \(s\) as an estimate for \(\sigma\), and then use the standard normal distribution. This method is based on the fact that for large samples, \(s\) is a fairly good approximation for \(\sigma\). Also, for large \(n\), the critical values for the Student's \(t\) distribution approach those of the standard normal distribution. Consider a random sample of size \(n=31\), with sample mean \(\bar{x}=45.2\) and sample standard deviation \(s=5.3\). (a) Compute \(90 \%, 95 \%\), and \(99 \%\) confidence intervals for \(\mu\) using Method 1 with a Student's \(t\) distribution. Round endpoints to two digits after the decimal. (b) Compute \(90 \%, 95 \%\), and \(99 \%\) confidence intervals for \(\mu\) using Method 2 with the standard normal distribution. Use \(s\) as an estimate for \(\sigma\). Round endpoints to two digits after the decimal. (c) Compare intervals for the two methods. Would you say that confidence intervals using a Student's \(t\) distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution? (d) Repeat parts (a) through (c) for a sample of size \(n=81\). With increased sample size, do the two methods give respective confidence intervals that are more similar?

Basic Computation: Confidence Interval for p Consider \(n=200\) binomial trials with \(r=80\) successes. (a) Cbeck Requirements Is it appropriate to use a normal distribution to approximate the \(\hat{p}\) distribution? (b) Find a \(95 \%\) confidence interval for the population proportion of successes \(p\). (c) Interpretation Explain the meaning of the confidence interval you computed.

Answer true or false. Explain your answer. A larger sample size produces a longer confidence interval for \(\mu\).

Psychology: Parental Sensitivity "Parental Sensitivity to Infant Cues: Similarities and Differences Between Mothers and Fathers" by M. V. Graham (Journal of Pediatric Nursing, Vol. 8, No. 6 ) reports a study of parental empathy for sensitivity cues and baby temperament (higher scores mean more empathy). Let \(x_{1}\) be a random variable that represents the score of a mother on an empathy test (as regards her baby). Let \(x_{2}\) be the empathy score of a father. A random sample of 32 mothers gave a sample mean of \(\bar{x}_{1}=69.44\). Another random sample of 32 fathers gave \(\bar{x}_{2}=59 .\) Assume that \(\sigma_{1}=11.69\) and \(\sigma_{2}=11.60\). (a) Check Requirements Which distribution, normal or Student's \(t\), do we use to approximate the \(\bar{x}_{1}-\bar{x}_{2}\) distribution? Explain. (b) Let \(\mu_{1}\) be the population mean of \(x_{1}\) and let \(\mu_{2}\) be the population mean of \(x_{2}\). Find a \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). (c) Interpretation Examine the confidence interval and explain what it means in the context of this problem. Does the confidence interval contain all positive, all negative, or both positive and negative numbers? What does this tell you about the relationship between average empathy scores for mothers compared with those for fathers at the \(99 \%\) confidence level?
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