Question

How do you compute confidence intervals?

Answer 1

Definition of confidence intervals:

A confidence interval is a type of interval estimate that is computed from the data that has been observed from a sample. It is an essential part of inferential statistics, upon which most market research is based.

To find the confidence intervals :

1. Find the number of samples $\left(n\right)$.
2. Calculate the mean (x) of the samples.

$mean\left(x\right)=\frac{\mathbf{s}\mathbf{u}\mathbf{m}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{t}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}\mathbf{}\mathbf{n}\mathbf{u}\mathbf{m}\mathbf{b}\mathbf{e}\mathbf{r}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{s}\mathbf{a}\mathbf{m}\mathbf{p}\mathbf{l}\mathbf{e}\mathbf{s}}{\mathbf{N}\mathbf{u}\mathbf{m}\mathbf{b}\mathbf{e}\mathbf{r}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{s}\mathbf{a}\mathbf{m}\mathbf{p}\mathbf{l}\mathbf{e}\mathbf{s}\mathbf{\left(}\mathbf{n}\mathbf{\right)}}$

3. Calculate the standard deviation $\left(SD\right)$.

$SD=\sqrt{\frac{{\displaystyle \stackrel{n}{\sum _{i=0}}{(x_i-x)}^2}}{\mathbf{n}}}$

where,

$SD=$the standard deviation of a sample

$n=$ number of samples

$x_i=$ each value of the sample

$x=$ number of samples

4. Decide the confidence interval that will be used.

95 percent and 99 percent confidence intervals are the most common choices in typical market research studies.

5. Find the $z$ value for the selected confidence interval.

$z=\frac{\left({\displaystyle \sum _{i=0}^n{x}_i-x}\right)}{\mathbf{S}\mathbf{D}}$

Confidence Interval	Z
80%	1.282
85%	1.440
90%	1.645
95%	1.960
99%	2.576
99.5%	2.807
99.9%	3.291

6. Calculate the formula by substituting the values

$x\pm z\frac{\mathrm{SD}}{\sqrt{\mathrm{n}}}$

where,

$SD=$the standard deviation of a sample

$n=$ number of samples

$x=$ number of samples

$z=$standard score

Hence, the computation of confidence intervals can be calculated by following these points.