CONFIDENCE

Returns a confidence interval.

Syntax:

CONFIDENCE(α, sd, size)

sd (> 0) is the (known) standard deviation of a normal distribution.

size is the size of a sample from that distribution.

α is the significance level (0 < α < 1), which determines the desired confidence level = (1 - α)*100%. Thus for example α = 0.05 gives a 95% confidence level.

CONFIDENCE returns a value that when added and subtracted from the sample mean gives an interval within which the population mean is expected to lie with the specified confidence level.

CONFIDENCE calculates NORMINV(1 - α/2; 0; 1) * sd / √(size)

Example:

CONFIDENCE(0.05,1,20)

returns approximately 0.438. With a 95% confidence level, the mean of the entire population lies within ±0.438 of the sample mean.

Application:

Imagine a company that makes soft drinks wants to estimate the average amount of soda a person drinks in a single sitting. They want to be 95% confident in their estimate. They can't survey the entire population, so they take a random sample of people.

Here's the data they have collected from a pilot study:

Sample Size (n): 50 people
Sample Mean ( $\overset{x}{ˉ}$ ): 16 ounces
Population Standard Deviation (σ): 3 ounces (This value is often an assumption based on prior knowledge or a large initial study).

The CONFIDENCE function would be used to calculate the margin of error for their survey results. The function's formula for a confidence interval for a mean (with a known population standard deviation) is:

$M a r g in o f E rror = Z_{α /2} * (\frac{σ}{n})$

Where:

Significance Level (α): This is the complement of the confidence level. For a 95% confidence level, the significance level is 5% or 0.05.
Z-value (Zα/2): This is a critical value from the standard normal distribution that corresponds to the significance level. For a 95% confidence level (two-tailed test with α/2=0.025), the Z-value is approximately 1.96.
Standard Deviation of the Normal Distribution (σ): 3 ounces.
Size of the Sample from that Distribution (n): 50 people.

Table of Inputs for the CONFIDENCE Function:

	Parameter	Value	Description
	A	B	C
1	Significance Level	0.05	For a 95% confidence level (1 - 0.95).
2	Standard Deviation	3 ounces	The assumed standard deviation of the population.
3	Sample Size	50 people	The number of individuals in the sample.

Now, let's perform the calculation:

Calculate the Standard Error:
- Standard Error = $\frac{σ}{n} = \frac{3}{50}$
Calculate the Margin of Error:

- Margin of Error = $Z_{α /2} \times Standard Error = 1.96 \times \frac{3}{50} \approx 0.832$

The CONFIDENCE function has now returned the margin of error: 0.832 ounces.

Constructing the Confidence Interval:

The confidence interval is the sample mean plus or minus the margin of error.

Lower Bound: Sample Mean - Margin of Error = 16 - 0.832 = 15.168 ounces
Upper Bound: Sample Mean + Margin of Error = 16 + 0.832 = 16.832 ounces

Conclusion:

Based on their survey and a 95% confidence level, the company can state with high confidence that the true average amount of soda consumed per person is somewhere between 15.17 and 16.83 ounces. This is a much more robust and statistically sound conclusion than simply stating the sample average of 16 ounces. It quantifies the uncertainty of the estimate, which is crucial for making informed business decisions.

Result for CONFIDENCE:

0.832