How to Calculate and Interpret Interval Estimates in Statistics
Estimation is the method of drawing conclusions regarding an unknown population parameter from a sample. An estimator is a statistic that is used to calculate the value of a parameter that is undefined. Intervals are statistical estimation approaches that employ survey data to generate value ranges that are likely to contain the population value of interest. Point estimates, on the other hand, are single-value estimates of a population value. Confidence intervals are the most well-known of the various forms of statistical intervals. Certain types of analyses and scenarios, on the other hand, necessitate the use of various ranges that provide different details.
In this article we will learn different interval estimation methods. We will learn prediction interval formula and many more along with examples.
Interval estimation in statistics is the computation of an interval, or set of values, within which a parameter. For example, the mean(average) of a population is most likely to be placed. The confidence coefficient is calculated by choosing intervals such that the parameter falls within them with a 95 or 99 percent probability. As a result, the intervals are referred to as confidence interval estimates. Upper and lower confidence limits are the end points of such an interval.
In comparison to point estimation, which is a single number, interval estimation uses sample data to measure an interval of possible (or probable) values for an unknown population parameter. In certain ways, a point estimate is the best estimate of the parameter based on a sample. Any point calculation, it should go without saying, isn't 100% correct. It's just a guess based on a single random sample. The point estimate would be expected to differ from sample to sample if repeated random samples were taken from the population.
Interval Estimate Formula, is Given Below
μ = \[\bar{x}\]\[\pm\] \[Z_{\frac{a}{2}}\frac{σ}{\sqrt{n}}\]
Where, \[\bar{x}\] = mean
\[Z_{\frac{a}{2}}\] = the confidence coefficient
α = Confidence Level
σ = Standard deviation
n = Sample size
Prediction Interval
Provided particular values of the independent variables, a prediction interval is a range that most likely includes the value of the dependent variable for a single new observation. We're estimating ranges for individual observations rather than the mean value for this sort of interval.
95 Prediction Interval
There is a 95% chance that a future observation will fall within the prediction interval if we collect a sample of observations and measure a 95 percent prediction interval based on that sample. On the other hand, there is a 5% chance that the next observation will not be contained within the interval. If we take 20 samples and determine a prediction interval for each, we can expect 19 of the intervals to contain a single future observation, while 1 of the intervals will not.
This interpretation of the prediction interval is depicted graphically in below diagram
[Image will be Uploaded Soon]
In the above diagram, upper and lower prediction interval bounds for 20 samples each containing 10 pH measurements drawn from the same population are plotted. There are also 20 individual future findings included in the storey. One of the twenty single future pH readings (circled in red) falls outside the associated prediction interval area. For a confidence level of 95 percent, this corresponds to the 5% of periods not anticipated to contain a single future observation.
The most popular use of prediction intervals is in regression statistics, but they can also be used for normally distributed results.
Point Estimate of the Population Mean
The most fundamental point and interval estimation process involves the estimation of a population mean. Assume you want to calculate the population mean,, for a quantitative variable. The sample mean x, where the value of x provides a point estimate of, can be computed using data from a simple random sample.
Since the sample mean is used as a point estimate of the population mean, some error can be expected due to the fact that the point estimate is computed using a sample, or subset of the population. The sampling error is the absolute value of the difference between the sample mean, x , and the population mean,μ, written |x̄ − μ|. A probability assumption about the size of the sampling error is included in interval estimation. The basis for such a statement is the sampling distribution of x .
Point estimation and interval estimation
In statistics, point estimators and interval estimators are the two most common types of estimators. Interval estimation is the polar opposite of point estimation. It yields a single value, while the latter yields a number of results. A point estimator is a statistic that is used to measure the value of a population's unknown parameter. When estimating a single statistic that will be the best approximation of the population's unknown parameter, it uses sample data.
Interval estimation, on the other hand, uses sample data to measure the range of potential values for a population's unknown parameter.
Example of Point Estimate is Given Below:
A single value of a statistic is a point estimate of a population parameter. The sample mean x, for example, is a point estimate of the population mean μ. The sample proportion p is a point estimate of the population proportion P, in the same way.
Common Methods of Finding Point Estimates
The method of point estimation entails using the value of a statistic derived from survey data to achieve the best approximation of the population's corresponding unknown parameter. The point estimators can be calculated using a variety of methods, each with its own set of properties.
Given Below are Two Methods to Find Point Estimates:
Method of Moments - It begins with known facts about a population, which are then applied to a sample of the population. The first step is to create equations that connect population moments to unknown parameters.
The next step is to select a population sample from which to estimate population moments. The sample mean of the population moments is used to solve the equations obtained in step one. This yields the most accurate estimation of unknown population parameters.
Maximum Likelihood Estimator - The maximum likelihood estimator is a point estimation tool that tries to find unknown parameters that increase the likelihood function. It compares data sets and finds the best fit for the data by using the values from a proven model.
For example, a study may be curious about the average weight of premature babies. Because measuring all premature babies in the population will be impractical, the researcher will take a sample from a single area. The researcher will use the maximum likelihood estimator to find the average weight of the entire population of pre-term babies based on the sample data since the weight of pre-term babies follows a normal distribution.
FAQs on Interval Estimation in Mathematics: Formula, Methods & Applications
1. What is interval estimation in statistics?
Interval estimation is a statistical method used to estimate an unknown population parameter by providing a range or an interval of plausible values. Unlike a point estimate, which gives a single value, an interval estimate communicates the degree of uncertainty. It is typically calculated as the point estimate plus or minus a margin of error.
2. What is the main difference between a point estimate and an interval estimate?
The main difference lies in the information they provide about a population parameter:
- A point estimate is a single number used to approximate the parameter. For example, stating that the average student exam score is 85.
- An interval estimate provides a range of values within which the parameter is likely to fall, along with a level of confidence. For example, stating that the average student exam score is between 82 and 88.
3. How is a confidence interval related to interval estimation?
A confidence interval is the most common and practical application of interval estimation. While interval estimation is the general theory of using a range to estimate a parameter, a confidence interval is a specific calculated range that is tied to a confidence level (e.g., 95% or 99%). The confidence level quantifies the long-term success rate of the method in capturing the true parameter.
4. How do you calculate an interval estimate for a population mean?
The general formula for an interval estimate is: Point Estimate ± Margin of Error. For a population mean, this translates to:
- Point Estimate: The sample mean (x̄).
- Margin of Error: This is calculated using the standard deviation of the population (or sample), the sample size (n), and a critical value (like a Z-score or t-score) from the standard distribution that corresponds to the desired confidence level.
5. What does a 95% confidence interval actually mean?
A 95% confidence interval has a specific interpretation that is often misunderstood. It does not mean there is a 95% probability that the true population parameter lies within your calculated interval. Instead, it means that if you were to repeat the sampling process numerous times and calculate a confidence interval for each sample, 95% of those calculated intervals would be expected to contain the true population parameter. It reflects the reliability of the estimation procedure itself.
6. Why is interval estimation often preferred over point estimation in scientific research?
Interval estimation is preferred because it provides a more complete and honest picture of the findings. A point estimate is precise but almost certainly wrong to some decimal place. An interval estimate, however, conveys two crucial pieces of information: a range of plausible values for the parameter and the level of precision of the estimate. A narrow interval indicates a more precise estimate, while a wide interval signals more uncertainty, which is vital for making informed decisions.
7. Can an interval estimate be wrong, and what factors affect its accuracy?
Yes, an interval estimate can be wrong. For instance, with a 95% confidence interval, there is a 5% chance that the interval you calculated does not capture the true population parameter. The accuracy and precision (width) of the interval are affected by:
- Sample Size: Larger samples lead to narrower, more precise intervals.
- Confidence Level: A higher confidence level (e.g., 99% vs. 95%) results in a wider, less precise interval.
- Data Variability: Data with a smaller standard deviation (less spread) produces a narrower interval.
8. What is the difference between a confidence interval and a prediction interval?
These two intervals serve different purposes. A confidence interval is used to estimate a population parameter, such as the average value for an entire population. A prediction interval is used to predict the value of a single future observation. Because a prediction interval must account for both the uncertainty in estimating the population parameter and the random variation of an individual data point, it is always wider than a confidence interval for the same data and confidence level.
9. Can you give a simple real-world example of interval estimation?
Imagine an e-commerce company wants to estimate the average delivery time for its packages. They take a sample of 200 orders and find the average delivery time is 3.1 days (the point estimate). Using interval estimation, they calculate a 95% confidence interval of [2.8 days, 3.4 days]. This means the company is 95% confident that the true average delivery time for all its packages lies somewhere between 2.8 and 3.4 days. This range is more useful for setting customer expectations than the single 3.1-day estimate.
