# How to use the CONFIDENCE.T function

**What is the CONFIDENCE.T function?**

The CONFIDENCE.T function calculates the confidence range for a population mean using a Student's t distribution. Use the CONFIDENCE.T function when you sample size is small (less than 30 observations). The t-distribution accounts for the increased uncertainty associated with small sample sizes and provides a more accurate estimate of the confidence interval.

The CONFIDENCE.NORM function is recommended if the sample size is larger than 30 observations.

#### Table of Contents

## 1. Introduction

**What is the confidence range for a population mean?**

A confidence interval around a sample mean provides a range of plausible values for the unknown population mean. It is calculated using the sample mean, standard error, and t-distribution, indicating the precision of the sample mean as an estimate of the actual population mean.

**What is a Student's t distribution?**

The Student's t distribution is a probability distribution that arises when estimating population means from samples, depending on the sample size and used to construct confidence intervals and in Student's t-tests.

The T distribution has heavier tails that accounts for the increased uncertainty associated with small sample sizes. The larger sample size the thinner tails, as size goes to infinity the t distribution becomes the normal distribution.

The image above shows the t distribution with one degree of freedom, the tails are very heavy. The normal distribution, which is also shown in the image above, has very thin tails.

## 2. CONFIDENCE.T Function Syntax

CONFIDENCE.T(*alpha,standard_dev,size*)

## 3. CONFIDENCE.T Function Arguments

alpha |
Required. The significance level. |

standard_dev |
Required. The standard deviation. |

size |
Required. The number of samples. |

**What is the significance level?**

The significance level, denoted α (alpha) , is the probability of that you want to use to calculate the confidence level. For example, a 95% significance level corresponds to a 0.05 alpha. 1 - 0.95 = 0.05

**What is standard deviation?**

Standard deviation tells you how far from the average values are spread out. Both charts above have numbers and an average plotted, they share the exact same average however, the numbers are not the same.

Chart A above shows that the values are more spread out than the values in chart B. Chart A has a standard deviation of 23.45256334, standard deviation for chart B is 5.207075606.

Standard deviation measures the amount of variation or dispersion of values in a dataset relative to the mean. It is calculated as the square root of the variance by taking

- each data point's deviation from the mean
- squaring it
- averaging the squares
- finally square rooting the result

Standard deviation = √(Σ(x - x̄)/n)

x̄ - arithmetic mean

n - count of values

## 4. CONFIDENCE.T function Example 1

**A marketing firm conducted a survey to estimate the average monthly expenditure of households in a city. They surveyed 20 randomly selected households and recorded their monthly expenditures. Calculate the 99% confidence interval for the mean monthly household expenditure in the city.**

**Average monthly expenditure : 1000**

**Standard deviation: 100**

**Size: 20**

The CONFIDENCE.T function is what we want to use since the sample size is only 20. It lets you calculate the interval that satisfies a given a significance level, standard deviation and a number representing the population size based on the t distribution.

The image above shows the arguments:

- alpha in cell C19 which is 1-0.99 = 0.01
- standard_dev in cell C20 = 100
- size in cell C21 = 20

Formula in cell C23:

The CONFIDENCE.T function returns approx 63.97 for the specified arguments above.We can now calculate the upper and lower limit by adding the confidence interval from cell C23 to the mean μ. I have done this in cell C25:

The lower limit is calculated by subtracting the confidence level to the mean μ. Cell 26:

In other words, there is a 99% probability that the average monthly expenditure is between approx. 936.03 and approx. 1063.97

## 5. CONFIDENCE.T function Example 2

**A nutritionist wants to estimate the average daily calorie intake of male adults in a certain region. They collect data on the daily calorie intake of 10 randomly selected male adults. Using this sample data in cell range E19:H21, calculate the 95% confidence interval for the mean daily calorie intake of male adults in the region. Here are the observations:
**

2194 | 2649 | 2407 | 2123 |

2896 | 2788 | 2064 | 2916 |

2249 | 2421 |

The CONFIDENCE.NORM function requires three arguments, the alpha, standard deviation and the size. The alpha is specified in question, 1 - 0.95 = 0.05

We can calculate the mean μ using the AVERAGE function. Formula in cell C24:

The average is calculated by adding all the measurements and then divide by the number of observations. The standard deviation is also easy to calculate in Excel. Formula in cell C25:

We now have all the arguments needed to calculate the interval that satisfies a given significance level at 80% (0.8).

- alpha in cell C19 which is 1-0.8 = 0.05
- standard_dev in cell C20 approx. equal to 321.48
- size in cell C21 = 10

Formula in cell C23:

The CONFIDENCE.T function returns 229.98 for the specified arguments above. We can now calculate the upper and lower limit by adding the confidence interval from cell C23 to the mean μ. I have done this in cell C25:

The lower limit is calculated by subtracting the confidence level to the mean μ. Cell 26:

In other words, there is a 95% probability that the mean daily calorie intake is between approx. 2240 and approx. 2701

## 6. CONFIDENCE.T function Example 3

**A study aims to determine the average reaction time of drivers when presented with a sudden obstacle on the road. The researchers recorded the reaction times of 15 randomly selected drivers in a controlled experiment. Calculate the 90% confidence interval for the mean reaction time of drivers in this population.**

**Average reaction time : 459 ms**

**Standard deviation: 78 ms**

**Size: 15
alpha: 1-0.9 = 0.1
**

Formula in cell C23:

The CONFIDENCE.T function returns approx 35.47 for the specified arguments above.We can now calculate the upper and lower limit by adding the confidence interval from cell C23 to the mean μ. I have done this in cell C25:

The lower limit is calculated by subtracting the confidence level to the mean μ. Cell 26:

In other words, there is a 90% probability that the average reaction time is between approx. 423.53 and approx. 494.47

### CONFIDENCE.T function not working

The CONFIDENCE.T function returns:

- #VALUE! error value if any argument is non-numeric.
- #NUM! error value if
*alpha*<= 0 (zero)*alpha*>= 1.*standard_dev*<= 0.*size*< 1.

- #DIV/0! error value if
*size*equals 1.

CONFIDENCE.T function truncates *size* argument to integers.

### Functions in 'Statistical' category

The CONFIDENCE.T function function is one of 73 functions in the 'Statistical' category.

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