October 17, 2020

Chapters

## What is a Confidence Interval?

A **confidence interval** reflects the extent of the uncertainty of a specific statistic. These intervals are mostly accompanied by the **margin of error**. The confidence interval helps you to determine how confident you can be that the results from a survey reflect the opinion or trend of the entire population. Confidence intervals are associated with confidence levels.

## Confidence Intervals Vs Confidence Levels

The two terms confidence intervals and confidence levels seem alike, however, there is a difference between these two terms. Confidence levels are represented as a percentage, for instance, the confidence level of this poll is 98%. It means that if you conduct the survey repeatedly, then 98% of the time the results of the poll will match the existing results.

Confidence intervals are represented in the form of numbers and they reflect the results of the survey. The confidence limits represent the two extreme values of the confidence interval that also reflect the range.

For instance, a survey is conducted in a locality to determine how much its residents spend on the grocery every month. After testing the statistics at a 95% confidence level, you get a confidence interval of (500,800). What does this interval reflect? Well, it means that the residents of that locality spend between $500 to $800 on groceries per month. You are 95% confident that the result of this survey is accurate.

## Margin of Error

A margin of error reflects by how much percentage points your result will deviate from the real value of the population. For instance, a confidence interval of 98% with a 5% margin of error means that your value will be within 5 percentage points of the real value of the population 98% of the time. The formula for computing the margin of error is given below:

**Margin of error =Critical value of the statistic x standard deviation**

In the next section, we will discuss the steps to find the confidence interval.

## Steps for Calculating the Confidence Interval

While solving the problems related to the confidence interval, you should follow the following steps:

### Step 1

Calculate the mean and standard deviation of the population. In some cases, they will be given in the problem, however, if they are not mentioned, you can calculate these values yourself.

The formula for calculating mean is:

The formula for computing the standard deviation of a sample is:

### Step 2

Determine the confidence interval you want for your sample and fetch the value of Z from the following table. In most cases, the confidence interval is 95% or 99% because these values indicate that the results are accurate.

Confidence Interval | Value of Z |
---|---|

80% | 1.282 |

85% | 1.440 |

90% | 1.645 |

95% | 1.960 |

99% | 2.576 |

99.5% | 2.807 |

### Step 3

Substitute the values of Z, the mean and standard deviation in the following formula to calculate the confidence interval.

Confidence Interval =

Here:

is the mean of the sample

s reflects the standard deviation of the population

Z is the value chosen from the table

n reflects the number of observations

Let us now proceed to solve some of the examples related to the confidence interval.

## Example 1

### Solution

Confidence Interval | Value of Z |
---|---|

80% | 1.282 |

85% | 1.440 |

90% | 1.645 |

95% | 1.960 |

99% | 2.576 |

99.5% | 2.807 |

## Example 2

### Solution

Confidence Interval | Value of Z |
---|---|

80% | 1.282 |

85% | 1.440 |

90% | 1.645 |

95% | 1.960 |

99% | 2.576 |

99.5% | 2.807 |

## Example 3

### Solution

Confidence Interval | Value of Z |
---|---|

80% | 1.282 |

85% | 1.440 |

90% | 1.645 |

95% | 1.960 |

99% | 2.576 |

99.5% | 2.807 |

## Example 4

### Solution

Confidence Interval | Value of Z |
---|---|

80% | 1.282 |

85% | 1.440 |

90% | 1.645 |

95% | 1.960 |

99% | 2.576 |

99.5% | 2.807 |

## Example 5

### Solution

Confidence Interval | Value of Z |
---|---|

80% | 1.282 |

85% | 1.440 |

90% | 1.645 |

95% | 1.960 |

99% | 2.576 |

99.5% | 2.807 |