91Ó°ÊÓ

Small Business: Bankruptcy 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

Understand the Problem

We need to determine the sample size required to estimate the proportion \( p \) of small businesses that declared Chapter 11 bankruptcy with a margin of error of \( 0.10 \) at a \( 95\% \) confidence level. We'll explore both scenarios: (a) without preliminary data and (b) using preliminary sample results.

02

Determine the Formula for Sample Size - Without Preliminary Data

Without any preliminary data, we assume maximum variability, meaning \( p = 0.5 \). Use the formula for sample size \( n \) in estimating proportions: \[ n = \left( \frac{Z^2 \cdot p \, (1-p)}{E^2} \right) \] where \( Z \) is the z-score for \( 95\% \) confidence level (\( Z = 1.96 \)), \( E = 0.10 \) is the margin of error.

03

Calculate Sample Size - Without Preliminary Data

Substitute \( p = 0.5 \), \( Z = 1.96 \), and \( E = 0.10 \) into the formula: \[ n = \left( \frac{(1.96)^2 \cdot 0.5 \, (1-0.5)}{(0.10)^2} \right) = \frac{3.8416 \cdot 0.25}{0.01} = 96.04 \] Rounding up, since sample size must be a whole number, we need \( n = 97 \) small businesses.

04

Determine the Formula for Sample Size - Using Preliminary Data

With preliminary data, \( \hat{p} = \frac{6}{38} \approx 0.158 \). Use the same formula: \[ n = \left( \frac{Z^2 \cdot \hat{p} \, (1-\hat{p})}{E^2} \right) \].

05

Calculate Sample Size - Using Preliminary Data

Substitute \( \hat{p} = 0.158 \), \( Z = 1.96 \), and \( E = 0.10 \) into the formula: \[ n = \left( \frac{(1.96)^2 \cdot 0.158 \, (1-0.158)}{(0.10)^2} \right) = \frac{3.8416 \cdot 0.133064}{0.01} \approx 51.00 \] Round up to get \( n = 52 \). Since 38 have already been sampled, \( 52 - 38 = 14 \) more businesses are needed.

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Key Concepts

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

Confidence Interval

The concept of a confidence interval is crucial in statistics as it helps us make informed estimates about a population parameter. A confidence interval gives a range of values within which the true population parameter is expected to fall. This is done with a certain level of confidence, typically expressed as a percentage. In our case, we are using a 95% confidence interval.

A 95% confidence level means that if we were to take 100 random samples from the population and compute a confidence interval for each sample, we would expect 95 of those intervals to contain the actual population parameter.

• A higher confidence level means more certainty about the estimate, but it also results in a wider interval.
• The width of the confidence interval is affected by the sample size; a larger sample size results in a narrower interval, assuming the same confidence level.

In our exercise, the confidence interval is used to ensure our estimate of the proportion of small businesses declaring bankruptcy is accurate within the specified range of the margin of error.

Proportion Estimation

Estimating a proportion involves determining the fraction of a population possessing a certain characteristic. Here, we are interested in finding out the proportion of small businesses that filed for Chapter 11 bankruptcy.

To estimate this proportion, we rely on a sample of data from the larger population. This is necessary since measuring the entire population may be impractical.

• The proportion is represented by the symbol \( p \) (population) or \( \hat{p} \) (sample).
• In initial scenarios where no data is available, we assume maximum uncertainty, which is \( p = 0.5 \).
• After collecting preliminary data, we update our proportion estimations, as demonstrated with \( \hat{p} = \frac{6}{38} \) for the initial sample in our exercise.

Using this, we apply statistical formulas to calculate the required sample size to achieve our desired confidence and accuracy levels.

Margin of Error

The margin of error is an important concept that measures the extent of the possible error in a sample-based estimate of a population parameter. It reflects the level of precision of our estimate.

In essence, it tells us how much the true population parameter can diverge from our sample estimate. A smaller margin means a more precise estimation, but might require a larger sample size.

• The formula for the margin of error \( E \) in the context of proportion estimation is given by \( E = Z \times \sqrt{\frac{p(1-p)}{n}} \), where \( Z \) is the Z-score corresponding to the desired confidence level, and \( n \) is the sample size.
• In our exercise, the desired margin of error is 0.10, meaning our estimate will not be off by more than 10% from the actual proportion.
• Adjusting the margin of error can affect both the required sample size and the confidence we can place in the results.

Thus, in any survey or experiment, carefully choosing an acceptable margin of error is crucial to balance resource constraints against the need for precision.