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Chapter 7: Problem 6

Estimate the minimum sample size needed to form a confidence interval for the proportion of a population that has a particular characteristic, meeting the criteria given. a. \(95 \%\) confidence, \(E=0.02\) b. \(99 \%\) confidence, \(E=0.02\) c. \(95 \%\) confidence, \(E=0.01\)

Short Answer

Expert verified
a) 2401, b) 4159, c) 9604.

Step by step solution

01

Understanding the Formula

To estimate the minimum sample size required for a proportion, we use the formula \( n = \left(\frac{Z^2 \cdot p \cdot (1-p)}{E^2}\right) \). Here, \( n \) is the sample size, \( Z \) is the Z-value from the confidence level, \( p \) is the estimated proportion (usually taken as 0.5 if not given), and \( E \) is the margin of error.
02

Identifying Z-values for Confidence Levels

For different confidence levels, we use specific Z-values: for \(95\%\) confidence, \( Z \approx 1.96 \); for \(99\%\) confidence, \( Z \approx 2.576 \). These values correspond to the number of standard deviations from the mean to enclose the desired proportion of the distribution.
03

Step 3a: Calculation for 95% Confidence, E=0.02

Substitute \( Z = 1.96 \), \( p = 0.5 \), and \( E = 0.02 \) into the formula: \[ n = \left(\frac{1.96^2 \cdot 0.5 \cdot (1-0.5)}{0.02^2}\right) \approx \left(\frac{3.8416 \cdot 0.25}{0.0004}\right) \approx 2401 \]. Thus, the minimum sample size needed is 2401.
04

Step 3b: Calculation for 99% Confidence, E=0.02

Substitute \( Z = 2.576 \), \( p = 0.5 \), and \( E = 0.02 \) into the formula: \[ n = \left(\frac{2.576^2 \cdot 0.5 \cdot (1-0.5)}{0.02^2}\right) \approx \left(\frac{6.635776 \cdot 0.25}{0.0004}\right) \approx 4158.6 \]. The minimum sample size needed is 4159 (rounded up to the nearest whole number).
05

Step 3c: Calculation for 95% Confidence, E=0.01

Substitute \( Z = 1.96 \), \( p = 0.5 \), and \( E = 0.01 \) into the formula: \[ n = \left(\frac{1.96^2 \cdot 0.5 \cdot (1-0.5)}{0.01^2}\right) \approx \left(\frac{3.8416 \cdot 0.25}{0.0001}\right) \approx 9604 \]. The minimum sample size needed is 9604.

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Key Concepts

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

Confidence Intervals
A confidence interval is like a range that tells us where a population parameter is likely to be. It's based on sample data and gives us a range where we believe the true value, such as a proportion or mean, lies. These intervals are important because they help us understand the potential error in our estimate. The width of a confidence interval can vary based on the sample size and the level of confidence we choose.
  • Confidence Level: This indicates how certain we are that the interval contains the true parameter value. Common levels are 95% or 99% confidence.
  • Variability and Sample Size: More sample variability or a smaller sample size can lead to a larger interval, so less precision.
  • Calculation: To calculate, you need a point estimate (like a sample mean or sample proportion) and then add and subtract a margin of error to get the range.
Remember, a 95% confidence interval means if we were to take 100 different samples and make an interval from each, we'd expect about 95 of those intervals to contain the true parameter value.
Margin of Error
The margin of error is a crucial part of understanding how much variation we're seeing in our data. It tells us the range of error around our sample estimate and is a way to express the uncertainty in our estimate.
  • Significance: A small margin of error indicates more confidence in our estimate, while a larger margin suggests more uncertainty.
  • Determining Factors: The size of the margin of error is influenced by the sample size, the variability in the data, and the level of confidence we choose.
  • Relation with Sample Size: Generally, a larger sample size will decrease the margin of error, making our estimate more precise.
Using the margin of error with the point estimate gives us the boundaries of the confidence interval, for example, if our estimated proportion is 0.5, with a margin of error of 0.02, our interval is 0.48 to 0.52.
Z-value
The Z-value plays a key role in constructing confidence intervals and determining sample size. It's a statistic that measures the number of standard deviations a data point is from the mean, within a normal distribution.
  • Application in Confidence Intervals: The Z-value corresponds to the desired confidence level. For example, for 95% confidence, we use a Z-value of approximately 1.96 because this value signifies the area under the normal curve spreading across those 95%.
  • Significance in Sample Size Calculation: When calculating sample size for estimating proportions, the Z-value helps determine how far our sample estimate might be from the true population parameter.
  • Variability with Confidence Levels: Different confidence levels have their unique Z-values; for instance, 99% confidence uses a Z-value around 2.576. Higher Z-values reflect greater confidence levels and imply a wider confidence interval, indicating more certainty that the interval contains the true parameter.
The chosen Z-value is essential to ensure that our confidence interval adequately reflects the uncertainty and precision we expect in our population estimate.

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Most popular questions from this chapter

The administration at a college wishes to estimate, to within two percentage points, the proportion of all its entering freshmen who graduate within four years, with \(90 \%\) confidence. Estimate the minimum size sample required.

A manufacturer of chokes for shotguns tests a choke by shooting 15 patterns at targets 40 yards away with a specified load of shot. The mean number of shot in a 30 -inch circle is 53.5 with standard deviation 1.6. Construct an \(80 \%\) confidence interval for the mean number of shot in a 30 -inch circle at 40 yards for this choke with the specified load. Assume a normal distribution of the number of shot in a 30 - inch circle at 40 yards for this choke.

Estimate the minimum sample size needed to form a confidence interval for the proportion of a population that has a particular characteristic, meeting the criteria given. a. \(p=0.81,95 \%\) confidence, \(E=0.02\) b. \(\quad p=0.81,99 \%\) confidence, \(E=0.02\) c. \(\quad p=0.81,95 \%\) confidence, \(E=0.01\)

A random sample of size 28 is drawn from a normal population. The summary statistics are \(\bar{x}-68.6\) and \(s=1.28\). a. Construct a \(95 \%\) confidence interval for the population mean \(\mu\). b. Construct a \(99.5 \%\) confidence interval for the population mean \(\mu\). c. Comment on why one interval is longer than the other.

To try to understand the reason for returned goods, the manager of a store examines the records on 40 products that were returned in the last year. Reasons were coded by 1 for "defective," 2 for "unsatisfactory," and 0 for all other reasons, with the results shown in the table. $$ \begin{array}{llllllllll} 0 & 2 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{array} $$ a. Give a point estimate of the proportion of all returns that are because of something wrong with the product, that is, either defective or performed unsatisfactorily. b. Assuming that the sample is sufficiently large, construct an \(80 \%\) confidence interval for the proportion of all returns that are because of something wrong with the product.
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