Examples of error bound in the following topics:
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- Working Backwards to find the Error Bound or the Sample Mean
- Subtract the error bound from the upper value of the confidence interval
- Suppose we know that a confidence interval is (67.18, 68.82) and we want to find the error bound.
- If we know the error bound: = 68.82 − 0.82 = 68
- If we don't know the error bound: = (67.18 + 68.82)/2 = 68
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- ( lower value,upper value ) = ( point estimate − error bound,point estimate + error bound )
- Formula 8.2: To find the error bound when you know the confidence interval
- error bound = upper value − point estimate OR error bound = (upper value − lower value)/2
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- Increasing the confidence level increases the error bound, making the confidence interval wider.
- Decreasing the confidence level decreases the error bound, making the confidence interval narrower.
- If we increase the sample size n to 100, we decrease the error bound.
- If we decrease the sample size n to 25, we increase the error bound.
- Decreasing the sample size causes the error bound to increase, making the confidence interval wider.
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- If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.
- The error bound formula for a population mean when the population standard deviation is known is $EBM = z_{\frac{\alpha }{2}} \cdot (\frac{\sigma }{\sqrt{n}})$
- The formula for sample size is $n = \frac{z^2\sigma ^2}{EBM^2}$ , found by solving the error bound formula for n
- A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.
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- Calculate the confidence interval and the error bound. i.
- Error Bound:
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- Would the error bound become larger or smaller?
- Using the same p' and n = 80, how would the error bound change if the confidence level were increased to 98%?
- If you decreased the allowable error bound, why would the minimum sample size increase (keeping the same level of confidence)?
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- Calculate the error bound.
- Calculate the error bound.
- Calculate the error bound.
- Calculate the error bound.
- Calculate the error bound.
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- Would the error bound become larger or smaller?
- Using the same mean, standard deviation and sample size, how would the error bound change if the confidence level were reduced to 90%?
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- Although they are often used interchangeably, the standard deviation and the standard error are slightly different.
- The standard error is the standard deviation of the sampling distribution of a statistic.
- However, the mean and standard deviation are descriptive statistics, whereas the mean and standard error describes bounds on a random sampling process.
- Despite the small difference in equations for the standard deviation and the standard error, this small difference changes the meaning of what is being reported from a description of the variation in measurements to a probabilistic statement about how the number of samples will provide a better bound on estimates of the population mean, in light of the central limit theorem.
- Standard error should decrease with larger sample sizes, as the estimate of the population mean improves.
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- This is due to the fact that the standard error of the mean is a biased estimator of the population standard error.
- However, while the mean and standard deviation are descriptive statistics, the mean and standard error describe bounds on a random sampling process.
- Despite the small difference in equations for the standard deviation and the standard error, this small difference changes the meaning of what is being reported from a description of the variation in measurements to a probabilistic statement about how the number of samples will provide a better bound on estimates of the population mean.
- If one survey has a standard error of $10,000 and the other has a standard error of $5,000, then the relative standard errors are 20% and 10% respectively.
- Paraphrase standard error, standard error of the mean, standard error correction and relative standard error.