Confidence Intervals

Introduction to Confidence Intervals

Let's start by looking at the terminology behind this important concept in Statistics.

The main reason you want to do an interval estimation through confidence intervals than a point estimation – a single statistic – is that sample results vary from sample to sample.

So, you need some measure of how much you can expect those results to change if you were to repeat your study. This expectation of variations in your statistic from sample to sample is measured by the margin of error.

Let's go back to the previous example.

Here, you are using your sample to estimate a range of values – a confidence interval – where there’s a likelihood to find the true value of the unknown parameter you’re interested in. This likelihood gives you that certain confidence in your results, and it is called confidence level.

Without further ado, let's see how to build a confidence interval.

Confidence Interval Formula

The terminology presented in the previous section actually gives you a clue to the elements needed to build a confidence interval. For example the formula for the confidence interval for the mean is:

\[CI=\overline{x}\pm z \frac{\sigma_s}{\sqrt{n}} \]

Here we can identity:

\(\overline{x}\): The sample mean.

\(z\): The confidence level.

\(\sigma_s\): The standard deviation of the sample.

\(n\): The sample size.

If you want to know more about samples, the sample mean, and the sample standard deviation, check our article named Sample Mean.

With these elements, you can build a confidence interval.

Types of Confidence Intervals

However, confidence intervals can mean several things.

The types of confidence intervals you will see below are:

• The confidence interval for population mean.

• The confidence interval for the difference of two means.

• The confidence interval for population proportion.

• The confidence interval for the difference of two proportions.

• The confidence interval for the slope of a regression model.

Confidence Interval for Population Mean

Let us say you take a sample \(a\) of a whole population \(A\). This sample \(a\) has a mean \(\overline{x_a}\). If the sampling has enough data and the survey is random, then the parameters of the sample will resemble the ones of the large population. The better the sampling method is, the better the mean of the sample will resemble the mean of the whole population.In this case, the confidence interval is the range \([x_1 - x_2]\) in the original sample \(a\), on which we have a probability value \(P\) to find the population mean.

This has another implication which is if you take another sample is very probable the mean of this sample will be in this range too. Let us make a numerical example.

The size of the sample will affect the confidence interval in the previous example. If the sample was only \(1000\) then the result will be \(0\pm0.051\).

Confidence Interval for the Difference of Two Means

Let us say you have two samples from two populations. Like the weight samples of a class in grade 8 in England and in grade 9 in Scotland. You want to find the difference between the means of both.This could be easy, calculate the mean in weight of the class in England \(w_E\) and subtract this from the mean of the class in Scotland \(w_S\). However, the samples are random, their means not resemble the mean of grade \(8\) in England and in grade \(9\) in Scotland. We have uncertainties about the possible rusult.In this case we have a formula to calculate the confidence interval. The mean of two different populations is defined as:

\[CI_p=(\overline{x_1} - \overline{x_2})+t\sqrt{\frac{sp^2}{n_1}+\frac{sp^2}{n_2}}\]

\(\overline{x_1}{,}\overline{x_2}\): The population means.

\(sp\): The pooled variance.

\(n_1, n_2\): The population of sample \(1\) and sample \(2\).

\(t\): The \(t\) critical value.

The pooled variance is calculated as follows:

\[sp=\sqrt{\dfrac{(n_1 - 1)\cdot s^2_1+n_2- 1)\cdot s_2^2 }{n_1 + n_2 -2}}\]

\(s_1{,}s_2\): are the variances of the samples.

Confidence Interval for Population Proportion

You have seen what happens with the confidence interval in a normal distribution. In these types of distributions, the values are continuous. However, there are other types of distributions, like the Binomial distribution. In this case, the values are the result of a Bernoulli-type experiment. In a Bernoulli-type, the results have only two outcomes.In these distributions, we can test a question.

\[p=Z\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\]

\(\hat{p}\): is the proportion or percentage.

\(Z\): is the value for the confidence level as in the table you used before.

\(n\): is the sample size.

Confidence Interval for the Difference of Two Proportions

Just as when you have the confidence interval of two means of two samples of two populations, this can exist also for proportions. In this case, you have two proportions obtained from samples.

The two samples survey the same question in populations \(A\) y \(B\), however their results are different \(\hat{p_1}\) and \(\hat{p_2}\). In this case, the confidence interval for the difference of two proportions is given by the next equation:

\[(\hat{p_1}-\hat{p_2})\pm Z \sqrt{\dfrac{\hat{p_1}(1-\hat{p_1})}{n_1}+\dfrac{\hat{p_2}(1-\hat{p_2})}{n_2} }\]

Confidence Interval for the Slope of a Regression Model

If you suspect that there might be a linear relationship between two variables, then you can construct a confidence interval for the slope of a regression model. Remember that you can use linear regression or the least-squares regression technique to create the line that best fits the data.

To learn more about this confidence interval, read our article on Confidence Intervals for the Slope of a Regression Model.

To learn how to draw conclusions from this type of confidence interval, read our article on Justifying Claims Based on the Confidence Interval for the Slope of a Regression Model.

Confidence Interval Interpretation

Again, a confidence interval is an interval with the likely values of a population parameter based on one or several random samples, with a \(c\%\) confidence level.

What the confidence level says is that the method used to create a particular confidence interval is successful in capturing the value of the actual population parameter approximately \(c\%\) of the time.

Beware: a confidence level of \(X\%\) does not mean the probability of the parameter being between the limits of the confidence interval is \(X\%\).

Again, a confidence level of \(c\%\) concerns the method used to produce the confidence interval.

So, the interpretation you should make of a confidence interval is that you can be \(c\%\) confident that the actual value of the parameter is included in the calculated interval.

The most common confidence levels are of \(90\%\), \(95\%\) and \(99\%\).

Considerations on the margin of error, confidence level and sample size

You could think by aiming for a narrow interval to estimate your parameter, you get closer to knowing its true value since it's more precise. It is more convenient and precise for you to know that you are meeting a friend in neighborhood \(X\), instead of in city \(Y\).

But in confidence intervals you should think the other way around: the smaller the width of the interval, the less sure you are that the true value of the parameter is in that interval. Although accuracy decreases, it is much safer to assume that the parameter is in city \(Y\), rather than in neighborhood \(X\) because the same city may contain other neighborhoods in which the parameter may be present.

This means

the higher the confidence level is, the wider the confidence interval.

A confidence interval with \(99\%\) confidence level is wider than one with a \(95\%\) confidence level, which is wider than one with a \(90\%\) confidence level, regarding the same situation.

Another thing you might notice in the formulas presented is that the sample size also affects the margin of error. In all the situations presented, the sample size \(n\) appears in the denominator of the standard error. Thus,

the larger the sample size, the narrower the confidence interval

(because the smaller the value of the standard error).

Example of Confidence Intervals

Let's end this article with two examples where we calculate the confidence interval of a mean and the confidence interval of two proportions.