Interval estimation is a statistical technique used to estimate population parameters by providing an interval (or range) of values within which the true parameter value is likely to lie, along with a level of confidence associated with the interval. This confidence level is expressed as a percentage (e.g., 95% confidence interval).
Steps for constructing a confidence interval are as follows:
- Estimate the Sample Statistic: Calculate the sample statistic (e.g., sample mean, sample proportion) from the data.
- Calculate the Standard Error: Determine the standard error of the sample statistic, which measures the variability of the estimate due to sampling.
- Select the Confidence Level: Choose the desired level of confidence for the interval (e.g., 95%).
- Compute the Margin of Error: The margin of error is based on the critical value from the standard normal distribution or t-distribution (depending on the sample size and assumptions). It is multiplied by the standard error.
- Construct the Interval: The confidence interval is constructed by adding and subtracting the margin of error from the sample statistic.
The resulting confidence interval provides a range of values within which we can be reasonably confident that the true population parameter lies. A 95% confidence interval, for example, means that if we were to take many random samples and construct a confidence interval for each sample, approximately 95% of those intervals would contain the true population parameter.
Interval estimation is widely used in inferential statistics to provide a measure of uncertainty and precision in estimating population parameters. It allows researchers to communicate the level of confidence associated with their estimates and aids in decision-making and hypothesis testing.
Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis usually represents the status quo or no effect, while the alternative hypothesis represents the effect or relationship of interest.
Hypothesis testing process involves the following steps:
- Formulate Hypotheses: Define the null and alternative hypotheses based on the research question and the nature of the problem being investigated.
- Select a Test Statistic: Choose an appropriate test statistic that measures the difference between the sample estimate and the hypothesized value under the null hypothesis.
- Set the Significance Level: Determine the significance level (α), which represents the probability of rejecting the null hypothesis when it is true. Commonly used significance levels are 0.05 (5%) and 0.01 (1%).
- Compute the Test Statistic: Calculate the test statistic using the sample data and the hypothesized value.
- Compare the Test Statistic to Critical Value(s): Determine the critical value(s) based on the significance level and the distribution of the test statistic. If the test statistic falls in the critical region (extreme tails of the distribution), we reject the null hypothesis.
- Make a Decision: Based on the comparison of the test statistic to critical value(s), either reject the null hypothesis in favor of the alternative hypothesis or fail to reject the null hypothesis.
Hypothesis testing is commonly used in various fields to assess the significance of relationships, compare groups, test the effectiveness of interventions, and draw conclusions based on sample data. The outcome of a hypothesis test provides evidence for or against the alternative hypothesis, leading to informed decision-making.