Problem 59 - Basic Statistics for Business and Economics with Student

Question

The proportion of public accountants who have changed companies within the last three years is to be estimated within 3 percent. The 95 percent level of confidence is to be used. A study conducted several years ago revealed that the percent of public accountants changing companies within three years was 21 a. To update this study, the files of how many public accountants should be studied? b. How many public accountants should be contacted if no previous estimates of the population proportion are available?

Step-by-Step Solution

Answer

a. 720 accountants; b. 1068 accountants.

Step 1: Understand the Problem

We need to determine the sample size necessary to estimate a population proportion with a specified margin of error and level of confidence. Two scenarios exist: one with a prior estimate of the proportion and one without.

Step 2: Formula for Sample Size

The sample size can be calculated using the formula:\[ n = \left( \frac{Z^2 \times p \times (1 - p)}{E^2} \right) \]where \( n \) is the sample size, \( Z \) is the Z-score, \( p \) is the estimated proportion, and \( E \) is the margin of error.

Step 3: Identify Given Values

For 95% confidence level, \( Z = 1.96 \). The margin of error \( E = 0.03 \). Given an earlier study, the estimated proportion \( p = 0.21 \).

Step 4: Calculate Sample Size with Prior Estimate

Substitute the known values into the formula:\[n = \left( \frac{1.96^2 \times 0.21 \times (1 - 0.21)}{0.03^2} \right) \approx 719.76\]Round up to the nearest whole number, so \( n = 720 \) accountants.

Step 5: Calculate Sample Size without Prior Estimate

When no prior estimate is available, use \( p = 0.5 \) for maximum variability.\[n = \left( \frac{1.96^2 \times 0.5 \times (1 - 0.5)}{0.03^2} \right) \approx 1067.11\]Round up to the nearest whole number, so \( n = 1068 \) accountants.

Key Concepts

Population Proportion

The term **population proportion** represents the fraction or percentage of the population that possesses a particular characteristic. In simple terms, it answers the question: how many out of a total group have a feature you're interested in?
For example, if we're looking specifically at the percentage of public accountants who've changed companies in the past three years, the population proportion would convey that exact figure.In statistical problems, we often use this known proportion to make predictions or calculate necessary figures like sample sizes. When calculating the sample size, having an estimated population proportion is crucial, as seen in the formula:\[ n = \left( \frac{Z^2 \times p \times (1 - p)}{E^2} \right) \]Here, \( p \) represents the estimate of the population proportion, serving as a starting point for calculations.
If we have a prior estimate, like 21% from an earlier study, this value becomes part of the equation. It's vital as it impacts the sample size, allowing for more accurate and tailored results.
Without an estimated proportion, we default to using \( p = 0.5 \) because it represents maximum variability, ensuring we account for every possible scenario.

Margin of Error

The **margin of error** details how close your sample results are likely to be to the actual population value. It conveys the range within which we expect the real percentage to lie, with a specific confidence level.● For example, in the exercise, you aim for a margin of error of 3% for the estimation of public accountants changing companies.
This means you want your sample's result to be within 3% of the actual proportion, with specified confidence.A smaller margin of error indicates a more precise estimate requires a larger sample size. Conversely, a larger margin results in lesser precision and thus, might reduce the necessary sample size.When used in the formula `E` (the margin of error) directly affects the sample size calculation:\[ n = \left( \frac{Z^2 \times p \times (1 - p)}{E^2} \right) \]
Here, \( E = 0.03 \), indicating a desire for high precision.
This precision choice impacts how much data is collected, ensuring estimates stay within the desired range. This aids in making results reliable enough to base decisions off of.

Confidence Level

The **confidence level** conveys the degree of certainty we have in the results of our sampling. Expressed as a percentage, it tells us how sure we are that the actual population parameter lies within our calculated range.Commonly used confidence levels are 90%, 95%, and 99%. Each corresponds to a different \( Z \) value in the formula:\[ n = \left( \frac{Z^2 \times p \times (1 - p)}{E^2} \right) \]For a 95% confidence level, \( Z \) equals approximately 1.96, which is what is used in the mentioned exercise.
This means if you repeated the study many times, approximately 95 out of 100 times, the actual population proportion would fall within this margin of error.A higher confidence level indicates greater reliability but also requires a larger sample size.
For instance, moving from 95% to 99% confidence level would increase \( Z \) to roughly 2.576, demanding a significantly larger sample.
This balance between confidence and sample size is crucial: higher confidence comes at the cost of additional data, time, and resources.