# Methods and formulas for Bootstrapping for 1-Sample Proportion

Select the method or formula of your choice.

## Proportion

### Notation

TermDescription
x number of events in the original sample
N number of trials in the original sample

## Average

### Formula

###### Note

Minitab does not take resamples from the original column of data. Instead, Minitab takes the resamples by randomly sampling from a binomial distribution. The parameters for the distribution are taken from the original sample.

### Notation

TermDescription
proportion of the ith resample
B number of resamples
N number of trials in the original sample

## Confidence interval

### Formula

Take a random sample from a binomial distribution with the following parameters:
• The number of trials is equal to the length of the original sample or the number of trials for summarized data.
• The event probability is equal to the proportion of events in the original sample or the number of events divided by the number of trials for summarized data.
• The number of samples taken is equal to the number of resamples.

Sort the proportion of the random samples in increasing order. x1 is the lowest number, xB is the highest number.

Lower bound: xl where =

Upper bound: xu where =

###### Note

For a one-sided case (only a lower bound or upper bound), use α instead of α/2.

When l or u are not integers, Minitab does a linear interpolation between the two numbers on either side of l or u. The formula is:

Xy + z(Xy+1 - Xy)

For example, if l = 5.25, the lower bound equals x5 + .25 (x6 - x5).

Minitab does not display the confidence interval when or .

### Notation

TermDescription
α 1 – confidence level/100
B number of resamples
Xy the yth row of data when the data are sorted from least to greatest
y the truncated value of l or u
zl-y or u - y
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