In preparing a report on the economy, we need to estimate...

In preparing a report on the economy, we need to estimate the percentage of businesses that plan to hire additional employees in the next 60 days. a) How many randomly selected employers must we contact in order to create an estimate in which we are $98 \%$ confident with a margin of error of $5 \%$ ? b) Suppose we want to reduce the margin of error to $3 \%$. What sample size will suffice? c) Why might it not be worth the effort to try to get an interval with a margin of error of only $1 \%$ ?

In preparing a report on the economy, we need to estimate the percentage of businesses that plan to hire additional employees in the next 60 days.
a) How many randomly selected employers must we contact in order to create an estimate in which we are $98 \%$ confident with a margin of error of $5 \%$ ?
b) Suppose we want to reduce the margin of error to $3 \%$. What sample size will suffice?
c) Why might it not be worth the effort to try to get an interval with a margin of error of only $1 \%$ ?

Key Concepts

Precision vs. Practicality Trade-off

This concept examines the balance between achieving a higher degree of precision in an estimate and the practical considerations such as time, cost, and effort involved in collecting a larger sample. As one seeks to reduce the margin of error (i.e., to increase precision), the required sample size grows, often exponentially. Decision-makers must assess whether the incremental improvement in estimation accuracy justifies the additional resources and effort.

Sample Size Calculation

Sample size calculation is the process of determining the number of observations needed to achieve a specified margin of error at a given confidence level. In the context of estimating proportions, this calculation typically involves the standard error of the proportion and the z-score associated with the chosen confidence level. A larger sample size generally reduces the margin of error, but the benefits must be weighed against additional costs and practical feasibility.

Confidence Level

Confidence level represents the degree of certainty that the true population parameter lies within the constructed confidence interval. It is expressed as a percentage and indicates the probability that the interval estimation process will capture the true parameter if repeated multiple times. In statistical decision-making, the confidence level is crucial because it defines the reliability and trustworthiness of the interval estimate.

Margin of Error

The margin of error is a measure that reflects the maximum expected difference between the true population parameter and a sample estimate. It indicates the precision of the estimate, with a smaller margin of error implying a more precise estimate. The margin of error directly influences the width of a confidence interval, and managing it involves a balance between the desired precision and practical constraints such as sample size and cost.

Transcript

Please give Ace some feedback