CONFIDENCE

See Also

Returns a value that you can use to construct a confidence interval for a population mean. The confidence interval is a range of values. Your sample mean, x, is at the center of this range and the range is x ± CONFIDENCE. For example, if x is the sample mean of delivery times for products ordered through the mail, x ± CONFIDENCE is a range of population means. For any population mean, μ₀, in this range, the probability of obtaining a sample mean further from μ₀ than x is greater than alpha; for any population mean, μ₀, not in this range, the probability of obtaining a sample mean further from μ₀ than x is less than alpha. In other words, assume that we use x, standard_dev, and size to construct a two-tailed test at significance level alpha of the hypothesis that the population mean is μ₀. Then we will not reject that hypothesis if μ₀ is in the confidence interval and will reject that hypothesis if μ₀ is not in the confidence interval. The confidence interval does not allow us to infer that there is probability 1 – alpha that our next package will take a delivery time that is in the confidence interval.

Syntax

CONFIDENCE(alpha,standard_dev,size)

Alpha    is the significance level used to compute the confidence level. The confidence level equals 100*(1 - alpha)%, or in other words, an alpha of 0.05 indicates a 95 percent confidence level.

Standard_dev    is the population standard deviation for the data range and is assumed to be known.

Size    is the sample size.

Remarks

• If any argument is nonnumeric, CONFIDENCE returns the #VALUE! error value.
• If alpha ≤ 0 or alpha ≥ 1, CONFIDENCE returns the #NUM! error value.
• If standard_dev ≤ 0, CONFIDENCE returns the #NUM! error value.
• If size is not an integer, it is truncated.
• If size < 1, CONFIDENCE returns the #NUM! error value.
• If we assume alpha equals 0.05, we need to calculate the area under the standard normal curve that equals (1 - alpha), or 95 percent. This value is ± 1.96. The confidence interval is therefore:

  

Example

Suppose we observe that, in our sample of 50 commuters, the average length of travel to work is 30 minutes with a population standard deviation of 2.5. With alpha = .05, CONFIDENCE(.05, 2.5, 50) returns 0.69291. The corresponding confidence interval is then 30 ± 0.69291 = approximately [29.3, 30.7]. For any population mean, μ₀, in this interval, the probability of obtaining a sample mean further from μ₀ than 30 is more than 0.05. Likewise, for any population mean, μ₀, outside this interval, the probability of obtaining a sample mean further from μ₀ than 30 is less than 0.05.

The example may be easier to understand if you copy it to a blank worksheet.

How?