# How to interpret Margin of Error Results?

How to interpret Margin of Error Results, In statistics, the margin of error is used to determine how accurate a population proportion or population mean estimate is.

When constructing confidence intervals for population parameters, we usually utilize the margin of error.

## How to interpret Margin of Error Results

For a population proportion and a population mean, the following examples demonstrate how to compute and understand margins of error.

### Example 1: Using the Margin of Error to Calculate the Population Proportion

To compute a confidence interval for a population proportion, use the following formula:

`Confidence Interval = p  +/-  z*(√p(1-p) / n)`

where:

p: sample proportion

z: the chosen z-value

n: sample size

The margin of error is the part of the equation that comes after the +/- sign:

`Margin of Error = z*(√p(1-p) / n)`

Let’s say we wish to estimate the percentage of citizens in a county that support a particular bill. We pick 100 residents at random and ask them about their opinions on the law.

The following are the outcomes.

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n = 100 sample size

p = 0.6 in support of the rule of law

Let’s say we want to establish a 95 percent confidence interval for the genuine proportion of county inhabitants who support the bill.

We determine the margin of error using the algorithm above:

`Margin of Error = z*(√p(1-p) / n)`

Margin of Error = 1.96*(√0.6(1-0.6) / 100)

Margin of Error =  0.09602

The 95 percent confidence interval can therefore be calculated as follows:

`Confidence Interval = p  +/-  z*(√p(1-p) / n)`

Confidence Interval = 0.6  +/-  0.09602

Confidence Interval = [0.50398, 0.69602]

The 95 percent confidence interval for the percentage of county inhabitants who support the legislation is [0.50398, 0.69602].

This indicates we have a 95% confidence level that the true percentage of residents who approve of the law is between 50.4 and 69.6 percent.

The percentage of inhabitants in favor of the bill in the sample was 60 percent, but we may generate a confidence interval by subtracting and adding the margin of error from this sample proportion.

This confidence interval provides a set of numbers that are very likely to contain the true percentage of county residents who support the law.

## Example 2: Using the Margin of Error to Calculate the Population Mean

To compute a confidence interval for a population mean, use the following formula.

`Confidence Interval = x  +/-  z*(s/√n)`

where:

x: sample mean

z: the z-critical value

s: sample standard deviation

n: sample size

The margin of error is the part of the equation that comes after the +/- sign:

`Margin of Error = z*(s/√n)`

Let’s say we wish to calculate the average weight of a dog population. We choose a random sample of dogs and record the following data.

This is only for the purpose of explaining and gathering some random data.

n = 40 sample size

x = 100 sample mean weight

s = 12 sample standard deviation

We determine the margin of error using the algorithm above:

`Margin of Error = z*(s/√n)`

Margin of Error = 1.96*(12/√40)

Margin of Error = 3.718839

We can then calculate the 95% confidence interval to be:

`Confidence Interval = x  +/-  z*(s/√n)`

Confidence Interval = 100 +/- 3.718839

Confidence Interval =[96.28116, 103.7188]

The mean weight of dolphins in this population has a 95 percent confidence interval of [96.28116, 103.7188].

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This indicates we have a 95% confidence level that the true mean weight of dogs in this population is between 96.3 and 103.7 pounds.

The average weight of the dogs in the sample was 100 pounds, but we can create a confidence interval by subtracting and adding the margin of error from the sample mean.

This confidence interval reflects a set of numbers that are very likely to include the true mean weight of dogs in this population.

We trust the concept is apparent to you.