margin of error

Examples of margin of error in the following topics:

• Analyzing Data

  • The margin of error is a statistic used to analyze data.
  • The margin of error can be described as an "absolute" quantity.
  • Therefore, the absolute margin of error is 5 people, but the "percent relative" margin of error is 10% (because 5 people are ten percent of 50 people).
  • The margin of error is a measure of how close the results are likely to be.
  • The FPC, factored into the calculation of the margin of error, has the effect of narrowing the margin of error.

• Caveat Emptor and the Gallup Poll

  • Readers should pay close attention to a poll's margin of error.
  • The margin of error statistic expresses the amount of random sampling error in a survey's results.
  • So in this case, the absolute margin of error is 5 people, but the "percent relative" margin of error is 10% (10% of 50 people is 5 people).
  • Also, if the 95% margin of error is given, one can find the 99% margin of error by increasing the reported margin of error by about 30%.
  • The larger the sample is, the smaller the margin of error is.

• Finding a sample size for a certain margin of error

  • How large of a sample is necessary to estimate the average systolic blood pressure with a margin of error of 4 mmHg using a 95% confidence level?
  • Recall that the margin of error is the part we add and subtract from the point estimate when computing a confidence interval.
  • The margin of error for a 95% confidence interval estimating a mean can be written as:
  • The challenge in this case is to find the sample size n so that this margin of error is less than or equal to 4, which we write as an inequality:
  • To estimate the necessary sample size for a maximum margin of error m, we set up an equation to represent this relationship:

• Proportion

  • The estimated standard error of p is therefore
  • Since the interval extends 0.045 in both directions, the margin of error is 0.045.
  • In terms of percent, between 47.5% and 56.5% of the voters favor the candidate and the margin of error is 4.5%.
  • Keep in mind that the margin of error of 4.5% is the margin of error for the percent favoring the candidate and not the margin of error for the difference between the percent favoring the candidate and the percent favoring the opponent.
  • The margin of error for the difference is 9%, twice the margin of error for the individual percent.

• Choosing a sample size when estimating a proportion

  • The task was to find a sample size n so that the sample mean would be within some margin of error m of the actual mean with a certain level of confidence.
  • For example, the margin of error for a point estimate using 95% confidence can be written as 1.96 × SE.
  • where ME represented the actual margin of error and z was chosen to correspond to the confidence level.
  • For a 95% confidence level, the value z corresponds to 1.96, and we can write the margin of error expression as follows:
  • What sample size does this estimate suggest we should use for a margin of error of 0.04 with 95% confidence?

• Estimates and Sample Size

  • As the confidence increases, the margin of error ($E$) increases.
  • To ensure that the margin of error is small, the confidence level would have to decrease.
  • As the sample size ($n$) increases, the margin of error decreases.
  • In this case, the margin of error, $E$, is found using the formula:
  • The larger the sample is, the smaller the margin of error is.

• Sample size and power exercises

  • (a) The standard error of ¯ x when s = 120 and (I) n = 25 or (II) n = 125.
  • (b) The margin of error of a confidence interval when the confidence level is (I) 90% or (II) 80%.
  • He would like to conduct another survey but have a margin of error of no more than $10 at a 99% confidence level.
  • We would like to calculate a 95% confidence interval for the average increase in reading speed with a margin of error of no more than 15%.
  • The higher the confidence level, the higher the corresponding margin of error.

• Perpetual vs. Periodic Counting

  • A company using the perpetual inventory system would have a book inventory that is exactly (within a small margin of error) the same as the physical (real) inventory.
  • Physical inventories are conducted at set time intervals; both cost of goods sold and the inventory are adjusted at the time of the physical inventory.
  • Many small businesses still only have a periodic system of inventory.
  • Perpetual inventory systems can still be vulnerable to errors due to overstatements (phantom inventory) or understatements (missing inventory) that occurs as a result of theft, breakage, scanning errors, or untracked inventory movements.
  • These errors lead to systematic errors in replenishment.

• Calculating the Sample Size n

  • If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.
  • The formula for sample size is $n = \frac{z^2\sigma ^2}{EBM^2}$ , found by solving the error bound formula for n
  • A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.
  • The population standard deviation for the age of Foothill College students is 15 years.
  • Therefore, 217 Foothill College students should be surveyed in order to be 95% confident that we are within 2 years of the true population mean age of Foothill College students.

• Bias

  • To better understand the outcome of experimental data, an estimate of the size of the systematic errors compared to the random errors should be considered.
  • Accuracy (or validity) is a measure of the systematic error.
  • If it is within the margin of error for the random errors, then it is most likely that the systematic errors are smaller than the random errors.
  • Calibration can eliminate this type of error.
  • Method Errors: This type of error many times results when you do not consider how to control an experiment.