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Magazine Chapter 7: Problem 3
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. \(\quad p=0.37,80 \%\) confidence, \(E=0.05\) b. \(\quad p=0.37,90 \%\) confidence, \(E=0.05\) c. \(\quad \rho=0.37,80 \%\) confidence, \(E=0.01\)
Short Answer
Expert verified
a. 235
b. 346
c. 14626
Step by step solution
01
Understanding the Problem
The problem asks us to estimate the minimum sample size needed to construct confidence intervals for a population proportion with specified confidence levels and margins of error. The provided proportions, confidence levels, and error margins need to be used in our calculations.
02
Formula for Sample Size Calculation
The sample size for estimating a population proportion can be calculated using the formula: \[ n = \left( \frac{Z_{\alpha/2}^2 \cdot p \cdot (1-p)}{E^2} \right) \] where \( n \) is the sample size, \( Z_{\alpha/2} \) is the Z-score corresponding to the desired confidence level, \( p \) is the estimated population proportion, and \( E \) is the margin of error.
03
Determining Z-scores
For an 80% confidence level, the Z-score \( Z_{\alpha/2} \) is approximately 1.28. For a 90% confidence level, \( Z_{\alpha/2} \) is approximately 1.645.
04
Calculating Sample Size for Part (a)
Use \( p = 0.37 \), \( E = 0.05 \), and \( Z_{\alpha/2} = 1.28 \). Substitute these values into the formula to find the sample size: \[ n = \left( \frac{1.28^2 \cdot 0.37 \cdot (1-0.37)}{0.05^2} \right) \] \[ n = \left( \frac{1.6384 \cdot 0.37 \cdot 0.63}{0.0025} \right) \] \[ n \approx 234.09 \] Rounding up, the minimum sample size needed is 235.
05
Calculating Sample Size for Part (b)
Use \( p = 0.37 \), \( E = 0.05 \), and \( Z_{\alpha/2} = 1.645 \). Substitute into the formula: \[ n = \left( \frac{1.645^2 \cdot 0.37 \cdot (1-0.37)}{0.05^2} \right) \] \[ n = \left( \frac{2.706025 \cdot 0.37 \cdot 0.63}{0.0025} \right) \] \[ n \approx 345.88 \] Rounding up, the minimum sample size needed is 346.
06
Calculating Sample Size for Part (c)
Use \( p = 0.37 \), \( E = 0.01 \), and \( Z_{\alpha/2} = 1.28 \). Substitute into the formula: \[ n = \left( \frac{1.28^2 \cdot 0.37 \cdot (1-0.37)}{0.01^2} \right) \] \[ n = \left( \frac{1.6384 \cdot 0.37 \cdot 0.63}{0.0001} \right) \] \[ n \approx 14625.9 \] Rounding up, the minimum sample size needed is 14626.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Confidence Interval
A confidence interval provides a range of values that is likely to contain the population parameter, such as a proportion or mean. When you construct a confidence interval, you are saying with a certain level of confidence (e.g., 80%, 90%) that the interval includes the true population parameter. This is useful because it gives you an estimate that accounts for sampling variability instead of relying on a single statistic.
Here's how it works:
Here's how it works:
- The "confidence level" tells you how sure you can be about the interval. For instance, an 80% confidence interval means if you were to take 100 different samples and compute an interval for each, about 80 of those intervals would contain the true parameter.
- The end points of the interval are determined by adding and subtracting the "margin of error" from the sample statistic.
Population Proportion
The population proportion is a parameter that represents the fraction or percentage of the total population that exhibits a particular characteristic. For example, if you're interested in the proportion of a town's voters who favor a candidate, this parameter will tell you that proportion in the whole town.
Here's why it's important:
Here's why it's important:
- To estimate this parameter, especially when the total population is very large or unknown, we use a sample proportion from a smaller, representative sample.
- This sample proportion then helps in constructing confidence intervals, letting us infer the likely range for the population proportion.
- Accurate estimation of population proportion is crucial for decision making, like determining the focus of marketing strategies or public policy.
Margin of Error
The margin of error tells us how much the observed value from a sample (like the sample proportion) can be expected to differ from the true population value. It is essentially the "plus-minus" value in a confidence interval.
In practice:
In practice:
- The margin of error gives a boundary around our sample estimate, within which the true population parameter is expected to lie, given the chosen confidence level.
- A smaller margin of error indicates that the sample result is likely to be closer to the true population value, but to achieve this often requires a larger sample size.
- The formula for the margin of error in a proportion can be expressed as: \( E = Z_{\alpha/2} \times \sqrt{\frac{p(1-p)}{n}} \)
Z-score
The Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. In the context of confidence intervals, the Z-score is used as a critical value that defines the boundaries of the confidence interval.
Some points to remember:
Some points to remember:
- The Z-score represents the number of standard deviations a data point is from the mean.
- In confidence intervals for population proportions, the Z-score is determined by the desired confidence level (e.g., 1.28 for 80% confidence, 1.645 for 90% confidence).
- Calculating the Z-score allows us to know just how "extreme" or "rare" a sample result is under a normal distribution assumption.
