Confidence Interval Estimation for The Average

Engineers have found the use of the confidence percentage discussed in the last section for estimating the average or average rather unfamiliar. They are more comfortable with the concept of the confidence interval. This term shows the range of the average having the degree of confidence (1 - a)%. The endpoints are referred to as the confidence limits. The formulas for the interval of the average estimation are for high- and low-volume samples, respectively:

Figure 5.3 shows an interpretation of the confidence interval for 13 samples from the same population with a known a. The different samples produce different values for X and, consequently, the interval spans are centered at different points. When the population σ is known, the confidence interval is the same for all samples, because all their confidence limits are derived from a. If the populationσis unknown, then the sample standard deviations (s) are used to calculate the confidence interval for each sample from Equation 5.7, and the span is different for different samples.

If the confidence limit was at 95% (or 2 = 2σaway from the average) then it is expected that the probability of at least one interval span falling outside the population average is 5%, or one out of 20 samples. Therefore, a sample whose average is outside the population average is considered unlikely to happen. In Figure 5.3, the unlikely sample is shown highlighted third from the top.

Example 5.6

A sample has the following characteristics: n = 81, sample average = 20, and standard deviation = 5. Find 95% and 99.9% confidence intervals，assuming that the population is normally distributed.

The sample is large enough to use 2 tables. From equation 5.6:

95% confidence (a = 0.25) = 20 士 1,960，5/9 = 20 土 1.09

99.9% confidence (a = 0.0005)= 20 土 3.290 • 5/9 = 20 土 1.83

Note that the confidence interval for 99.9% is almost double the one for 95%.

Example 5.7

For a sample of the following values, 2.6, 2.1, 2.4f 2.5, 2.7,2.2,2.3,2.4, and 1.9, find the confidence interval of the population average, assuming that it is normal, for 90%, 95%, and 99.9% confidence.

For the sample data: n = 2\ sample average X = 2.34, and sample standard deviation s = 0.25. Using the t distribution with and

Equation (5.7):

90% confidence (a = 0.05) = 2.34 土 1.860.0.25/3

= 2.34 ±0.16 (2.18— 2.5)

95 % confidence (a= 0.025) = 2.34 ± 2.306.0.25/3

= 2.34 ±0.19 (2.15-2.53)

99.9% confidence (a = 0.0005) = 2.34 土 5.041.0.25/3

= 2.34 ±0.42 (2.76-1.92)

In every case, the sample point 1.9 falls outside the lower confidence limit, making it an unusual event. At 99.9% confidence, the point has a probability of less than 0.005.