The problem with using a point estimate is that although it is the single best guess you can make about the value of a population parameter, it is also usually wrong. Interval estimate overcomes this problem using interval estimation technique which is based on point estimate and margin of error.
Point Estimation
Point estimation involves calculating a single value from sample data to estimate a population parameter. The examples of point estimation are: (i) Estimating the population mean using the sample mean and (ii) Estimating the population proportion using the sample proportion. The common point estimators are:
- Sample mean $\overline{x}$ for population mean ($\mu$).
- Sample proportion ($\hat{p}$​) for population proportion ($P$).
- Sample variance ($s^2$) for population variance ($\sigma^2$).
Interval Estimation
Interval estimation involves calculating a range of values (set of values: an interval) from sample data to estimate a population parameter. The range constructed has a specified level of confidence. The Components of an interval are:
- Confidence level: The probability that the true population parameter lies within the interval.
- Margin of error: The maximum allowable error (difference between the point estimate and the true population parameter).
The common confidence intervals for the population mean are:
- Confidence interval for a large sample (or known population standard deviation):
$\overline{x} \pm Z_{\alpha/2} \frac{s}{\sqrt{n}}$ - Confidence interval for small sample (or unknown population standard deviation):
$\overline{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$ - Confidence interval for the population proportion
$\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p} {1-\hat{p}}}{n}}$
Advantages of Interval Estimation
- A major advantage of using interval estimation is that you provide a range of values with a known probability of capturing the population parameter (e.g. if you obtain from SPSS a 95% confidence interval you can claim to have 95% confidence that it will include the true population parameter.
- An interval estimate (i.e., confidence intervals) also helps one not to be so confident that the population value is exactly equal to the single-point estimate. That is, it makes us more careful in interpreting our data and helps keep us in proper perspective.
- Perhaps the best thing to do is to provide both the point estimate and the interval estimate. For example, our best estimate of the population mean is the value of $32,640 (the point estimate) and our 95% confidence interval is $30,913.71 to $34,366.29.
- By the way, note that the bigger your sample size, the more narrow the confidence interval will be.
- Remember to include many participants in your research study if you want narrow (i.e., exact) confidence intervals.
In essence, interval estimation is a game-changer in the field of statistics. Interval estimation, acknowledges the uncertainty inherent in data, providing a range of probable values (interval estimates) instead of a single (point estimate), potentially misleading, point estimate. By incorporating it into the statistical analysis, one can gain a more realistic understanding of the data and can make more informed decisions based on evidence, not just a single number.
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