Page 171 - Data Science class 11
P. 171
In statistics, a confidence interval is an educated guess about some characteristic of the population. A confidence
interval contains an initial estimate plus or minus a margin of error (the amount by which you expect your results to
vary, if a different sample were taken).
A confidence interval shows how much uncertainty there is with any particular statistics. Confidence intervals are
often used with a margin of error. Confidence intervals give us a range of plausible values for some unknown value
based on results from a sample.
A confidence interval is a range of values that is likely to contain an unknown population parameter.The confidence
level represents the theoretical ability of the analysis to produce accurate intervals if you are able to assess many
intervals and you know the value of the population parameter.
Increasing the confidence level increases the error bound, making the confidence interval wider. Decreasing the
confidence level decreases the error bound, making the confidence interval narrower.
In statistics, it is used to indicate the probability, with which the estimation of the location of a statistical parameter
(e.g. an arithmetic mean) in a sample survey is also true for the population. In surveys, confidence levels of 90/95/99%
are frequently used.
The width of a confidence interval depends upon two things:
1. Diversity within the population of interest: If all the population values were nearly the same, then we will have
less variation. The estimate will be close to the actual population. Therefore, the confidence interval, in this case,
will be small. But a more diverse population will lead to a more diverse sample. Different samples taken from the
same population will vary more. There will be not be much assurance if the mean of the sample will be closer to
the population mean. As such, the confidence interval will be large. So greater diversity in the population leads to
a wider confidence interval.
2. The sample size affects the width of the confidence interval: With a small sample, we do not have much
reference to base our inference. Small samples will be different from one another, causing a wider confidence
interval. Whereas, in larger sample size, the effect of a few unusual values is evened out by the other values in the
sample. Larger samples will be much similar. The effective sampling error is minimised with larger samples. Larger
samples produce more information and estimates that a researcher is confident about, leading to a narrower
confidence interval.
Determining a Confidence Interval
If you want to be more than 95% confident about your results, you need to add and subtract more than about two
standard errors. For example, to be 99% confident, you would add and subtract about two and a half standard errors
to obtain your margin of error (2.58 to be exact).
What does 95% confidence level mean?
A 95% confidence interval is a range of values that you can be 95% certain. It contains the true mean of the population.
The mean for large samples has much more precision as compared to small samples, so the confidence interval is
quite narrow when computed from a large sample.
With a 95 percent confidence interval, you have a 5 percent chance of being wrong. With a 90 percent confidence
interval, you have a 10 percent chance of being wrong.
What does 99 confidence level mean?
A 99 percent confidence interval would be wider than a 95 percent confidence interval (for example, plus or minus 4.5
percent instead of 3.5 percent). In the vernacular, "we are 99% certain (confidence level) that most of these samples
(confidence intervals) contain the true population parameter."
Randomisation 169

