what is the exact meaning of 98% confidence interval? please explain with an example if possible.
In conducting a test of significance or hypothesis test there are two numbers that are easy to get confused.
These numbers are easily confused because they are both numbers between zero and one, and are in fact probabilities.
One number is called the p-value of the test statistic. The other number of interest is the level of significance, or alpha.
We will examine these two probabilities and determine the difference between them.
Alpha – The Level of Significance
The number alpha is the threshold value that we measure p values against. It tells us how extreme observed results
must be in order to reject the null hypothesis of a significance test.
The value of alpha is associated to the confidence level of our test. The following lists some levels of confidence with their related values of alpha:
For results with a 90% level of confidence, the value of alpha is 1 - 0.90 = 0.10.
For results with a 95% level of confidence, the value of alpha is 1 - 0.95 = 0.05.
For results with a 99% level of confidence, the value of alpha is 1 - 0.99 = 0.01.
And in general, for results with a C% level of confidence, the value of alpha is 1 – C/100.
Although in theory and practice many numbers can be used for alpha, the most commonly used is 0.05.
The reason for this both because consensus shows that this level is appropriate in many cases, and historically it has been accepted as the standard.
However, there are many situations when a smaller value of alpha should be used. There is not a single value of alpha that always determines statistical significance.
The alpha value gives us the probability of a type I error. Type I errors occur when we reject a null hypothesis that is actually true.
Thus, in the long run, for a test with level of significance of 0.05 = 1/20, a true null hypothesis will be rejected one out of every 20 times.
The other number that is part of a test of significance is a p-value. A p-value is also a probability, but it comes from a different source than alpha. Every test statistic has a corresponding probability or p-value. This value is the probability that the observed statistic occurred by chance alone, assuming that the null hypothesis is true.
Since there are a number of different test statistics, there are a number of different ways to find a p-value. For some cases we need to know the probability distribution of the population.
The p-value of the test statistic is a way of saying how extreme that statistic is for our sample data. The smaller the p-value, the more unlikely the observed sample.
To determine if an observed outcome is statistically significant, we compare the values of alpha and the p -value. There are two possibilities that emerge:
The p-value is less than or equal to alpha. In this case we reject the null hypothesis. When this happens we say that the result is statistically significant. In other words, we are reasonably sure that there is something besides chance alone that gave us an observed sample.
The p-value is greater than alpha. In this case we fail to reject the null hypothesis. When this happens we say that the result is not statistically significant. In other words, we are reasonably sure that our observed data can be explained by chance alone.
The implication of the above is that the smaller the value of alpha is, the more difficult it is to claim that a result is statistically significant. On the other hand, the larger the value of alpha is the easier is it to claim that a result is statistically significant. Coupled with this, however, is the higher probability that what we observed can be attributed to chance
Hope this helps
As the name suggests its an Interval around a Point estimate(Estimating a population parameter is why we ended up even having this term). As the name suggests again, its an estimate of a population parameter.
You estimate a mean, add a ± error and construct an interval. So its a range.
First sample - Range - 90-110
Second sample - Range - 100-115
Third sample - Range - 80-105
What 95% confidence interval states is that, when you do this experiment by sampling the same data multiple times or estimating the population parameter using sampling, 95 out 100 times the true population parameter will lie within the range of your estimate.
That is when you repeat the process to estimate the population parameter, you will be confident to say 95 out of 100 times you were able to estimate the population parameter while there will be 5 times out of 100 where the population parameter can be estimated.
Now read the above post of actual tests, interpretation of P values and alpha.
In the end its the cost of having number of incorrect estimates while determining population parameter that decides the CI and CL.
If it was 99% it simply means only 1 out of 100 of your sampling experiment will have a false estimate of population parameter with a ± error.
thanks for taking trouble of typing such a great answer. it’s really helpful.
thanks for the answer.