The results of a confidence interval and significance test should agree as long as: (Select all that apply)
- We are making inferences about means
- We are making inferences about proportions
- The significance test is two-sided
- Both are conducted on the same data set
D. Both are conducted on the same data set
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A. We are making inferences about means
C. The significance test is two-sided
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What is a Confidence Interval?
If you’re taking a statistics test in school, your teacher has likely shared with you the word “confidence interval” or a statistic called “p-value.” But what is a confidence interval?
A confidence interval is just an easy way to show how accurate an estimate of the parameter (usually for research purposes, something like your IQ) could be. This comes down to the accuracy of your data and when the sample size or population being researched is small, how precise you want the confidence interval to be.
Confidence intervals are often calculated to show the range within which we can be 95% confident that the true value falls; they can also be used to indicate how well a continuous variable conforms to normality assumptions.
The formula for confidence interval is (CI) = ‾X ± Z(S ÷ √n)
- x̄: Sample Mean
- z: Confidence Coefficient
- ơ: Population Standard Deviation
- n: Sample Size
What is a Significance Test?
A significance test is a procedure used to assess whether your results differ from the null hypothesis (which may be the case when there is an effect in your population) and they are used to state the probability that if you re-performed the same experiment on the same population, your results would be exactly the same.
Technically, a significance test is an assessment of whether or not a difference in a data set is due to chance or not.
The Four Types of Significance Tests
There are four types of significance tests. They include:
1. Student’s T-Test or T-Test
The Student’s T-test is a relatively simple yet important statistical test used to examine whether two sets of data are significantly different from each other. The test determines if the difference between means is bigger than the t-value for a given experiment or if it is less than the t-value for a given experiment (this produces an even smaller difference in which case we say there is no significant difference).
2. F-test or Variance Ratio Test
The F-test is used to compare the squared and the Standard F are used to compare the variance between two sets of data and look at how much difference there is between them. This is important in business because we often want to know whether the two sets of numbers are similar or not.
3. Fisher’s Z-Test or Z-Test
The Z-test is a test used to calculate the probability of drawing a certain number of positive results, given that the null hypothesis is true. This test is used extensively when dealing with small samples and large score variance.
4. Chi Square Test or X2-Test
The chi-square test assesses the goodness of fit between a set of data and a theoretical distribution (the null hypothesis). In other words, it helps determine whether there is an effect in the population such as an outlier or not.
What is Statistical Significance
Statistical significance is a test used to determine whether or not your results differ from the null hypothesis. If you believe that your results are significant, then you will conclude that there is a bias or an error in your sample. You can use statistical significance to state the probability that if you re-performed the same experiment on the same population, your results would be exactly the same.
The null hypothesis is the proposition that there is no effect in the population such as an outlier or between a sample and the observed data. The alternative hypothesis is the proposition that there is some effect on the population (your results are different from these observations). The test of statistical significance determines whether you can reject the null hypothesis or not.
Process of Significance Testing
To carry out a significance test you should follow these steps:
1. State the Research Hypothesis
An established association between two variables is stated in a study hypothesis. It can be described in broad strokes or with regard to quantity and direction. For instance,
General: The number of trainees who find jobs is correlated with the length of the job training program.
Direction: The rate of trainee job placement increases with the length of the training program.
Magnitude: Longer training programs will result in twice as many graduates finding employment.
General: Gender affects the compensation of graduate assistants.
2. State the Null Hypothesis
The null hypothesis is the proposition that there is no effect on the population. It could be phrased to fit a specific situation. For instance,
General: There is no relationship between the length of training and placement rates of trainees.
Direction: Length of training does not affect placement rate.
Magnitude: The longer a trainee’s training program, the less likely it will be that they will find work when they graduate.
3. Select a probability of error level
The significance level is the probability of rejecting the null hypothesis when it is true. A common convention is to use α=0.05, which means that there is a 5 percent chance you will reject the null hypothesis in error. It seems conservative to many researchers because it provides a high barrier for rejecting any null hypothesis (you must be very confident in your results before you conclude that they are significantly different).
4. Select and compute the test for statistical significance
The test of statistical significance determines whether you can reject the null hypothesis or not. It is a self-contained formula that gives you the quantity of information needed to decide if your results differ from the prediction made in the null hypothesis.
The test of statistical significance will only be significant if all conditions are met. If one or more conditions are not met, then your result will probably not be significant and it is best to accept that a possible effect exists.
5. Interpret the results
The results of a significance test are either “statistically significant” or “not statistically significant.” If the test results are not statistically significant, then it is best to conclude that you have found no effect.
Reasons to Use Confidence Intervals Over Significance Tests
1. Because confidence intervals don’t use the word “significance,” they avoid the false perception that it means “important.”
Confidence intervals serve as a reminder that all estimates are prone to error and that there is no such thing as an accurate estimate.
2. Confidence intervals offer more details than a test of statistical significance.
The test of significance would also show that the sample estimate was not statistically different from 0 at the 5% level if, at the 95 percent confidence level, a confidence interval for an effect includes 0.
3. A sense of the magnitude of any effect is given by the confidence interval. The data in a confidence interval are expressed in the applicable descriptive statistic (percentage, correlation, regression, etc.).
When a test of significance is applied in isolation, this effect size data is not present. The range estimate for the disparity between the average incomes of men and women in our income example was $6509 to $7700. This provides an idea of the approximate size of the difference as well as the error margin associated with any such difference.