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Magazine Chapter 9: Problem 16
Construct a confidence interval of the population proportion at the given level of confidence. \(x=80, n=200,98 \%\) confidence
Short Answer
Expert verified
The 98% confidence interval for the population proportion is (0.3194, 0.4806).
Step by step solution
01
Calculate the sample proportion
The sample proportion (\(\bar{p}\)) is calculated by dividing the number of successes (\(x\)) by the sample size (\(n\)). Therefore, \(\bar{p} = \frac{x}{n} = \frac{80}{200} = 0.4\)
02
Determine the critical value
For a 98% confidence level, the critical value (\(z^*\)) corresponds to the z-score that captures the middle 98% of the standard normal distribution. This value is approximately 2.33.
03
Compute the standard error
The standard error (SE) of the sample proportion is calculated using the formula \( SE = \sqrt{\frac{\bar{p}(1 - \bar{p})}{n}} \). Substituting the values, \( SE = \sqrt{\frac{0.4 \times 0.6}{200}} \approx 0.0346 \)
04
Calculate the margin of error
The margin of error (ME) is given by \( ME = z^* \times SE \). Substituting the values, \( ME = 2.33 \times 0.0346 \approx 0.0806 \)
05
Construct the confidence interval
Finally, the confidence interval is constructed using the formula \( \bar{p} \pm ME \). Substituting the values, the interval is \( 0.4 \pm 0.0806 \), resulting in the interval \( (0.3194, 0.4806) \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sample Proportion
The sample proportion is a way to estimate how often an event occurs within a sample, and it's a crucial first step in any confidence interval calculation. In our example, the sample proportion (\(\bar{p}\)) is calculated by dividing the number of successes (\(x\)) by the sample size (\(n\)). This gives us a number representing a success rate within our sample.
For the given problem: \[ \bar{p} = \frac{x}{n} = \frac{80}{200} = 0.4. \]
Meaning, 40% of the samples were successful. This value helps us understand the ratio of successes in the total sample surveyed.
For the given problem: \[ \bar{p} = \frac{x}{n} = \frac{80}{200} = 0.4. \]
Meaning, 40% of the samples were successful. This value helps us understand the ratio of successes in the total sample surveyed.
Critical Value
The critical value (\(z^*\)) is an essential part of constructing confidence intervals, representing how many standard deviations a point is from the mean for a chosen confidence level.
For a 98% confidence level, you'll often find critical values in statistical tables or by using statistical software. For a 98% confidence level, the critical value is approximately 2.33, meaning that 98% of data points fall within 2.33 standard deviations from the mean in a standard normal distribution. This value ensures the interval we create will be narrow enough to be meaningful, but wide enough to be accurate.
For a 98% confidence level, you'll often find critical values in statistical tables or by using statistical software. For a 98% confidence level, the critical value is approximately 2.33, meaning that 98% of data points fall within 2.33 standard deviations from the mean in a standard normal distribution. This value ensures the interval we create will be narrow enough to be meaningful, but wide enough to be accurate.
Standard Error
The standard error (SE) measures the variability of the sample proportion. It's a way to understand how much the sample proportion will fluctuate from sample to sample.
To calculate SE for a sample proportion, we use: \[ SE = \sqrt{\frac{\bar{p}(1 - \bar{p})}{n}} \] Substituting our known values: \[ SE = \sqrt{\frac{0.4 \times 0.6}{200}} \approx 0.0346 \].
Thus, the standard error is approximately 0.0346, reflecting a reasonable level of uncertainty or spread around the sample proportion.
To calculate SE for a sample proportion, we use: \[ SE = \sqrt{\frac{\bar{p}(1 - \bar{p})}{n}} \] Substituting our known values: \[ SE = \sqrt{\frac{0.4 \times 0.6}{200}} \approx 0.0346 \].
Thus, the standard error is approximately 0.0346, reflecting a reasonable level of uncertainty or spread around the sample proportion.
Margin of Error
The margin of error (ME) provides an interval around the sample proportion, helping us understand the range where the true population proportion likely lies.
To calculate the margin of error, multiply the critical value by the standard error: \[ ME = z^* \times SE \].
In our problem: \[ ME = 2.33 \times 0.0346 \approx 0.0806. \].
This margin of error expands our sample proportion by an approximate range of 0.0806 on either side, giving us the buffer needed to create an accurate confidence interval.
To calculate the margin of error, multiply the critical value by the standard error: \[ ME = z^* \times SE \].
In our problem: \[ ME = 2.33 \times 0.0346 \approx 0.0806. \].
This margin of error expands our sample proportion by an approximate range of 0.0806 on either side, giving us the buffer needed to create an accurate confidence interval.
Standard Normal Distribution
A standard normal distribution is a special case of a normal distribution with a mean of 0 and a standard deviation of 1. It is used to find the critical values for confidence intervals.
Any normal distribution can be converted to this standard form using Z-scores, making it easier to work with and look up probabilities. In the context of our problem, the critical value (2.33) was derived from this distribution to encompass the central 98% of the data, ensuring our confidence interval will correctly capture the population parameter. This standardization helps streamline the calculations and allows for easier reference and comparison.
Any normal distribution can be converted to this standard form using Z-scores, making it easier to work with and look up probabilities. In the context of our problem, the critical value (2.33) was derived from this distribution to encompass the central 98% of the data, ensuring our confidence interval will correctly capture the population parameter. This standardization helps streamline the calculations and allows for easier reference and comparison.
