To determine the proportion of taxpayers who prefer filing...

Name: Problem 83, Chapter 8: Applied Statistics in Business and Economics
Uploaded: 2021-02-25T01:57:11-08:00
Duration: 34 s
Channel: Ahmed Genedy
Description: To determine the proportion of taxpayers who prefer filing tax returns electronically, a survey of 600 taxpayers was conducted. Calculate the margin of error used to estimate this proportion. What assumptions are required to find the margin of error?

To determine the proportion of taxpayers who prefer filing tax returns electronically, a survey of 600 taxpayers was conducted. Calculate the margin of error used to estimate this proportion. What assumptions are required to find the margin of error?

Key Concepts

Random Sampling and Independence Assumptions

It is assumed that the sample is drawn randomly from the population, ensuring that each observation is independent of the others. This random sampling assumption is crucial, as it underpins the validity of the statistical inferences made from the sample, including the use of the normal approximation for the distribution of the sample proportion.

Standard Error

The standard error measures the variability of the sample proportion and is calculated using the formula ?[p(1-p)/n]. It reflects the precision of the sample estimate; a smaller standard error indicates a more precise estimate. When the true proportion is unknown, a conservative estimate (such as the worst-case scenario at p = 0.5) might be used to ensure that the margin of error is not underestimated.

Normal Approximation

The normal approximation is based on the central limit theorem, which states that with a large enough sample size, the distribution of the sample proportion will approximate a normal distribution regardless of the underlying distribution. This approximation is considered valid when both np and n(1-p) are sufficiently large (commonly at least 10), ensuring that the sampling distribution is adequately symmetric for constructing a reliable confidence interval.

Margin of Error

The margin of error quantifies the maximum expected difference between the sample estimate and the true population parameter. It is calculated by multiplying a critical value (often from the normal distribution) by the standard error of the estimate, and it provides a range within which the true proportion is expected to lie under a given confidence level.

Sampling Distribution of the Proportion

The sampling distribution of a sample proportion is the probability distribution that describes how the proportion measured in a sample would vary if different samples were taken from the same population. With a sufficiently large sample size, this distribution approximates a normal distribution, which is essential for constructing confidence intervals.

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