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Chapter 8: Problem 69

A consumer agency wants to estimate the proportion of all drivers who wear seat belts while driving. What is the most conservative estimate of the minimum sample size that would limit the margin of error to within \(.03\) of the population proportion for a \(99 \%\) confidence interval?

Short Answer

Expert verified
Based on the calculations, the conservative estimate of minimum sample size is approximately 752.

Step by step solution

01

Recognize the Key Values

The values given are: Margin of Error (E) = 0.03, Confidence Level (Z) = 99%, hence Z-value (\(Z_{\alpha/2}\)) = 2.57. Unknown is: standard deviation (p) of the population proportion, hence assumed as 0.5 for conservative estimate.
02

Use Formula for Sample Size

The formula for calculating sample size is: \(n = \left(\frac{Z_{\alpha/2}\cdot p\cdot(1-p)}{E}\right)^2.\) Substitute the known values into the formula: \( n = \left(\frac{2.57\cdot0.5\cdot0.5}{0.03}\right)^2\)
03

Calculate Sample Size

Perform the calculation in the formula to find the sample size. For fractional results, round up to the nearest integer, because size of a sample cannot be fractional.

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Key Concepts

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

Sample Size Estimation
When conducting a survey or sampling in statistics, determining the appropriate sample size is crucial. It ensures that the results are both reliable and applicable to the entire population. Sample size estimation involves calculating how many observations or data points are needed to achieve a certain level of accuracy in your results.
There are several factors that influence sample size estimation:
  • Confidence Level: This reflects how confident you are that the population parameter lies within the confidence interval. In our example, we are using a 99% confidence level.
  • Margin of Error: This is the maximum amount by which the estimated parameter is expected to differ from the true population parameter. We want our margin of error to be no more than 0.03 in this exercise.
  • Population Proportion (p): For conservative estimates, we assume that this is 0.5, as we do not have prior information.
By substituting these values into the formula \(n = \left(\frac{Z_{\alpha/2}\cdot p\cdot(1-p)}{E}\right)^2\), where \(Z_{\alpha/2}\) is the Z-value for our specific confidence level, we can calculate the minimum sample size required.
Margin of Error
The margin of error is a crucial concept in statistics, representing the range within which we expect the true population parameter to lie, given our sample statistic. It tells us how much uncertainty is included in the results we obtain from a sample.
Several elements affect the margin of error:
  • Sample Size: A larger sample size generally results in a smaller margin of error. This is because the sample mean tends to converge on the population mean as the sample size increases.
  • Confidence Level: Higher confidence levels will increase the margin of error, making the confidence interval wider.
  • Population Proportion (p): The variability of the data affects the margin of error. When a conservative estimate of 0.5 is used, it maximizes the error margin.
In our exercise, we aim for a margin of error of 0.03. By understanding and calculating the margin of error, we gain insight into the possible variance between our sample estimate and the actual population proportion.
Population Proportion
In statistics, the population proportion is the fraction of the total population that has a certain attribute. It's denoted by \(p\), and understanding it is vital when performing confidence interval calculations.
The calculation of the population proportion in a sample is straightforward:
  • Formula: The proportion \(\hat{p}\) is simply the number of successes (e.g., people wearing seat belts) divided by the total sample size \(n\).
  • Conservative Estimate: When estimating sample size without prior data, assuming that \(p = 0.5\) provides the most conservative perspective. This maxes the variability at \(p\cdot(1-p) = 0.25\).
Using this proportion helps in creating confidence intervals, which give us a range for where the true population proportion most likely falls. For accurate data representation, identifying the population proportion correctly is essential.
Z-value in Statistics
The Z-value (or Z-score) in statistics is a measure that describes a data point's relation to the mean of a group of values. It's expressed in terms of standard deviations from the mean, and it is widely used in calculating confidence intervals.
For determining confidence intervals, the Z-value is crucial because:
  • Confidence Level: Each confidence level corresponds to a specific Z-value. For instance, a 99% confidence level corresponds to a Z-value of 2.57.
  • Standard Normal Distribution: The Z-value shows how many standard deviations an observation is from the mean in a standard normal distribution. It helps in determining the width of the confidence interval.
In the exercise, the Z-value of 2.57 was used to determine the sample size needed for a 99% confidence interval. Understanding and selecting the appropriate Z-value is crucial for accurate confidence interval estimation.

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Most popular questions from this chapter

Briefly explain how the width of a confidence interval decreases with an increase in the sample size. Give an example.

What assumption(s) must hold true to use the normal distribution to make a confidence interval for the population proportion, \(p\) ?

A hospital administration wants to estimate the mean time spent by patients waiting for treatment at the emergency room. The waiting times (in minutes) recorded for a random sample of 35 such patients are given below. The population standard deviation is not known. \(\begin{array}{rrrrrrr}30 & 7 & 68 & 76 & 47 & 60 & 51 \\ 64 & 25 & 35 & 29 & 30 & 35 & 62 \\ 96 & 104 & 58 & 32 & 32 & 102 & 27 \\ 45 & 11 & 64 & 62 & 72 & 39 & 92 \\ 84 & 47 & 12 & 33 & 55 & 84 & 36\end{array}\) Construct a \(99 \%\) confidence interval for the corresponding population mean.

A random sample of 34 participants in a Zumba dance class had their heart rates measured before and after a moderate 10 -minute workout. The following data correspond to the increase in each individual's heart rate (in beats per minute): \(\begin{array}{llllllllllll}59 & 70 & 57 & 42 & 57 & 59 & 41 & 54 & 44 & 36 & 59 & 61 \\ 52 & 42 & 41 & 32 & 60 & 54 & 52 & 53 & 51 & 47 & 62 & 62 \\ 44 & 69 & 50 & 37 & 50 & 54 & 48 & 52 & 61 & 45 & & \end{array}\) a. What is the point estimate of the corresponding population mean? b. Make a \(98 \%\) confidence interval for the average increase in a person's heart rate after a moderate 10 -minute Zumba workout.

You want to estimate the proportion of students at your college who hold off- campus (part-time or full-time) jobs. Briefly explain how you will make such an estimate. Collect data from 40 students at your college on whether or not they hold off-campus jobs. Then calculate the proportion of students in this sample who hold offcampus jobs. Using this information, estimate the population proportion. Select your own confidence level.
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