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The National Council of Small Businesses is interested in the proportion of small businesses that declared Chapter 11 bankruptcy last year. Since there are so many small businesses, the National Council intends to estimate the proportion from a random sample. Let \(p\) be the proportion of small businesses that declared Chapter 11 bankruptcy last year. (a) If no preliminary sample is taken to estimate \(p\), how large a sample is necessary to be \(95 \%\) sure that a point estimate \(\hat{p}\) will be within a distance of \(0.10\) from \(p\) ? (b) In a preliminary random sample of 38 small businesses, it was found that six had declared Chapter 11 bankruptcy. How many more small businesses should be included in the sample to be \(95 \%\) sure that a point estimate \(\hat{p}\) will be within a distance of \(0.10\) from \(p ?\)

Step by step solution

01

Determine Margin of Error Formula

To find the necessary sample size when no preliminary sample is taken, we start with the formula for the margin of error (E) in proportion estimates: \[ E = z \times \sqrt{\frac{p(1-p)}{n}} \] where \(z\) is the z-score for the confidence level, \(p\) is the proportion (unknown, so assumed as 0.5 for maximum variability), and \(n\) is the sample size. We want \(E = 0.10\) with a 95% confidence level, which corresponds to a \(z\)-score of approximately 1.96.

02

Calculate Sample Size Without Preliminary Sample

We rearrange the margin of error formula to solve for \(n\): \[ n = \left(\frac{z^2 \cdot p(1-p)}{E^2}\right) \] Plugging in the values \(z = 1.96\), \(p = 0.5\), and \(E = 0.10\), we have: \[ n = \left(\frac{1.96^2 \cdot 0.5 \cdot 0.5}{0.10^2}\right) = \left(\frac{3.8416 \times 0.25}{0.01}\right) \approx 96.04 \] Since \(n\) must be a whole number, we round up to 97. Thus, 97 businesses are needed.

03

Calculate Sample Proportion from Preliminary Sample

In the preliminary sample, 6 out of 38 businesses declared bankruptcy. Thus, the sample proportion \(\hat{p}\) is: \[ \hat{p} = \frac{6}{38} \approx 0.1579 \] This value will be used to recalculate the required sample size with existing data.

04

Calculate Additional Sample Size Required Using Preliminary Data

Using the margin of error formula with \(\hat{p} = 0.1579\) from the preliminary sample: \[ n = \left(\frac{z^2 \cdot \hat{p}(1-\hat{p})}{E^2}\right) = \left(\frac{1.96^2 \cdot 0.1579 \times (1-0.1579)}{0.10^2}\right) \] \[ n = \left(\frac{3.8416 \cdot 0.1579 \times 0.8421}{0.01}\right) \approx 51.6 \] Rounding up, we need 52 total businesses. Since the preliminary sample was 38, we need an additional \(52 - 38 = 14\) businesses.

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Key Concepts

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

Sample Size Calculation

When trying to estimate a population parameter like the proportion of small businesses that declared bankruptcy, calculating the correct sample size is crucial. This sample size ensures that the point estimate from our study is reliable. The margin of error is a key part of this calculation, telling us how much our sample estimate might differ from the actual population value. When no preliminary sample data is available, a common strategy is to assume a proportion of 0.5. This conservative approach aims to maximize the required sample size, ensuring ample data to reflect the population accurately.
To calculate the sample size, the formula used is:

• \[ n = \left(\frac{z^2 \cdot p(1-p)}{E^2} \right) \] This formula helps determine the minimum sample size needed for a given confidence level and margin of error.
• Here, \(z\) is the z-score related to our confidence level—1.96 for a 95% confidence interval.
• \(E\) is our maximum tolerable error, or margin of error (such as 0.10).

By substituting these values into the formula, and assuming a worst-case scenario proportion, a researcher can determine the needed sample size even without preliminary data.

Confidence Intervals

Confidence intervals are essential in statistics, providing a range within which we expect the true population parameter to fall. For proportion estimation, this interval offers both a lower and an upper bound for the calculated proportion. The confidence level, often expressed as a percentage (like 95%), indicates how confident we are that the true proportion lies within this range.
The interplay between the confidence level and the margin of error is vital:

• A higher confidence level means a wider confidence interval, providing more certainty at the cost of precision.
• A smaller margin of error results in a narrower interval, focusing the estimation but potentially decreasing confidence.

In practice, when we estimate proportions, the standard approach is to calculate a 95% confidence interval, giving a satisfactory balance between precision and reliability. Regardless of the data, always remember that a confidence interval does not give absolute certainty but rather a probabilistic estimate.

Proportion Estimation

Proportion estimation is a fundamental part of statistical analysis, particularly when assessing the characteristics of a population with a large number of elements. This becomes crucial when trying to understand a feature of interest, like bankruptcy in small businesses, within a broader group.

• The sample proportion, \(\hat{p}\), is calculated as the number of successful outcomes divided by the total sample size.
• For example, observing 6 bankruptcies out of 38 businesses gives a sample proportion of \(\hat{p} = \frac{6}{38} \approx 0.1579\).

This sample proportion serves as the best point estimate of the true unknown population proportion. Adjusting the sample size by factoring in this sample proportion optimizes the estimate, often leading to a smaller required sample size than the conservative 0.5 initial assumption. Hence, using preliminary sample data refines not just the precision of the estimate but also aids in a more efficient study design by possibly reducing the number of samples needed.