10.5: The Summary of Confidence Intervals

We learned altogether 4 procedures for constructing confidence intervals to estimate 3 different parameters.

So how do we know when and which procedure to use?

First, we must determine which parameter is being estimated by looking for the key words such as

• Mean/average when estimating the population mean.
• Proportion/percentage when estimating the population proportion.
• Variance/standard deviation when estimating the population variance.

Next, we determine the procedure in the following way:

We always use

• The \(\chi^2\) procedure to estimate variances.
• The \(Z\) procedure to estimate proportions.

To estimate a sample mean, we need to check whether the population standard deviation is known

• If it is known, then we use One Mean Z Procedure.
• If it is not known, then we use One Mean T Procedure.

Table \(\PageIndex{1.1}\): Summary of confidence interval procedures in regard to purposes and assumptions.

Procedure	One mean \(Z\)-interval	One mean \(T\)-interval	One proportion \(Z\)-interval	One variance \(\chi^2\)-interval
Purpose	To find a confidence interval for a population mean, \(\mu\)	To find a confidence interval for a population mean, \(\mu\)	To find a confidence interval for a population proportion,  \(p\)	To find a confidence interval for a population variance, \(\sigma^2\)
Assumptions	Simple random sample Normal population or large sample \(\sigma\) is known	Simple random sample Normal population or large sample \(\sigma\) is unknown	Simple random sample \(np\) and \(n(1-p)\) are both 10 or greater.	Simple random sample Normal population

Remember - before applying the procedure we have to verify all the assumptions!

So before starting the procedure we must have already figured out the unknown parameter along with its point estimate.

Table \(\PageIndex{1.2}\): Summary of confidence interval procedures in regard to unknown parameters and point estimates.

Procedure	One mean \(Z\)-interval	One mean \(T\)-interval	One proportion \(Z\)-interval	One variance \(\chi^2\)-interval
Unknown Parameter	population mean, \(\mu\)	population mean, \(\mu\)	population proportion,\(p\)	population variance, \(\sigma^2\)
Point Estimate	sample mean, \(\bar{x}_i\)	sample mean, \(\bar{x}_i\)	sample proportion, \(\hat{p}_i\)	sample variance, \(s^2_i\)

ow confident are we that the point estimate equals exactly the unknown parameter? Zero percent confident. In other words, there are zero chances that the point estimate equals the parameter.

Next, we are going to construct an interval estimate so we need a new confidence level. Usually, it is given in the problem, if not then 95% is the most common confidence level in applications. If you want to be more confident make it 99%, if you want to be less confident then make it 90%. Whatever confidence level you choose, the value \(\alpha\) is such that \((1-\alpha)100\%\) is equal to the confidence level.

Table \(\PageIndex{1.3}\): Summary of confidence interval procedures in regard to identifying confidence level and alpha.

Procedure	One mean \(Z\)-interval	One mean \(T\)-interval	One proportion \(Z\)-interval	One variance \(\chi^2\)-interval
Confidence level	95% is the most common confidence level in applications If you want to be more confident then make it 99%, if you want to be less confident then make it 90%.
\(\alpha\)	\(CL=(1-\alpha)100\%\)

Next, we determine the critical values. Depending on the procedure you may have to do it in several steps. In Z procedures, we only need \(\alpha/2\), in T procedures, we also need the degrees of freedom, and in \(\chi^2\) procedures, we need both, \(\alpha/2\) and \(1-\alpha/2\) and the degrees of freedom.

Table \(\PageIndex{1.4}\): Summary of confidence interval procedures in regard to identifying the critical value(s).

Procedure	One mean \(Z\)-interval	One mean \(T\)-interval	One proportion \(Z\)-interval	One variance \(\chi^2\)-interval
\(\alpha/2\) or \(1-\alpha/2\)	\(\alpha/2\)	\(\alpha/2\)	\(\alpha/2\)	\(\alpha/2\), \(1-\alpha/2\)
Critical Value(s)	\(\pm z_{\alpha/2}\)	\(\pm t_{\alpha/2}\) \(df=n-1\)	\(\pm z_{\alpha/2}\)	\(\chi^2_{1-\alpha/2}\), \(\chi^2_{\alpha/2}\) \(df=n-1\)

Next, we construct the confidence interval by figuring out the lower and upper bounds. For one mean and one proportion procedures we do that by finding the margin of error first and then adding it to and subtracting it from the point estimate. For the one variance procedure, we compute the lower and upper bounds directly using the formulas due to the fact that the interval is not symmetric around the point estimate and therefore there is no margin of error.

Table \(\PageIndex{1.5}\): Summary of confidence interval procedures in regard to identifying the lower and upper bounds and margins of error when applicable.

Procedure	One mean \(Z\)-interval	One mean \(T\)-interval	One proportion \(Z\)-interval	One variance \(\chi^2\)-interval
Margin of Error	\(z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}\)	\(t_{\alpha/2}\dfrac{s}{\sqrt{n}}\)	\(z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}\)	n/a
Lower Bound	\(\bar{x}_i-ME\)	\(\bar{x}_i-ME\)	\(\hat{p}_i-ME\)	\(\dfrac{s^2_i(n-1)}{\chi^2_{\alpha/2}}\)
Upper Bound	\(\bar{x}_i+ME\)	\(\bar{x}_i+ME\)	\(\hat{p}_i+ME\)	\(\dfrac{s^2_i(n-1)}{\chi^2_{1-\alpha/2}}\)

Finally, we interpret the interval by writing or saying the following:

“We are (1-alpha)100% confident that the population proportion is between the LB and UB.”

We always include the units in the answer. Also, it is not uncommon to convert the proportion to percentages, and variance to standard deviation by taking the square root of the bounds.

For One Mean and One Proportion Z Procedures it is also possible to derive the formula that relates the size of the sample to the margin of error, so that we can determine the sample size necessary to achieve a certain level of accuracy.

Table \(\PageIndex{1.6}\): Summary of confidence interval procedures in regard to identifying the sample size to achieve the desired margin of error when applicable.

Procedure	One mean \(Z\)-interval	One mean \(T\)-interval	One proportion \(Z\)-interval	One variance \(\chi^2\)-interval
Desired margin of error	\(E_0\)	n/a	\(E_0\)	n/a
The sample size	\(n>\left(z_{\alpha/2}\cdot\dfrac{\sigma}{E_0}\right)^2\)	n/a	\(n>\left(\dfrac{z_{\alpha/2}}{2E_0}\right)^2\)	n/a

All four procedures can be put in one table in the following way:

Table \(\PageIndex{1.7}\): Summary of all confidence interval procedures.

Procedure	One mean \(Z\)-interval	One mean \(T\)-interval	One proportion \(Z\)-interval	One variance \(\chi^2\)-interval
Purpose	To find a confidence interval for a population mean, \(\mu\)	To find a confidence interval for a population mean, \(\mu\)	To find a confidence interval for a population proportion,  \(p\)	To find a confidence interval for a population variance, \(\sigma^2\)
Assumptions	Simple random sample Normal population or large sample \(\sigma\) is known	Simple random sample Normal population or large sample \(\sigma\) is unknown	Simple random sample \(np\) and \(n(1-p)\) are both 10 or greater.	Simple random sample Normal population
Point estimate	sample mean, \(\bar{x}_i\)	sample mean, \(\bar{x}_i\)	sample proportion, \(\hat{p}_i\)	sample variance, \(s^2_i\)
Confidence Level \((1-\alpha)100\%\)	\(\alpha/2\)	\(\alpha/2\)	\(\alpha/2\)	\(\alpha/2\), \(1-\alpha/2\)
Critical value(s)	\(\pm z_{\alpha/2}\)	\(\pm t_{\alpha/2}\) \(df=n-1\)	\(\pm z_{\alpha/2}\)	\(\chi^2_{1-\alpha/2}\), \(\chi^2_{\alpha/2}\) \(df=n-1\)
Margin of Error	\(z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}\)	\(t_{\alpha/2}\dfrac{s}{\sqrt{n}}\)	\(z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}\)	n/a
Lower Bound	\(\bar{x}_i-ME\)	\(\bar{x}_i-ME\)	\(\hat{p}_i-ME\)	\(\dfrac{s^2_i(n-1)}{\chi^2_{\alpha/2}}\)
Upper Bound	\(\bar{x}_i+ME\)	\(\bar{x}_i+ME\)	\(\hat{p}_i+ME\)	\(\dfrac{s^2_i(n-1)}{\chi^2_{1-\alpha/2}}\)
Interpretation	We are \((1-\alpha)100\%\) confident that the unknown population parameter is between LB [units] and UB [units].