# Examples of the use of the /-Distribution for Sample and Population Averages

### Example 5.1

A manufacturing line produces resistors in a normal process with an average value of 500 ohms. A Sample of nine resistors were taken from yesterday^ production, with sample average = 540 ohms and sample standard deviation = 60. Does the sample indicate that the production process was out of control yesterday?

Solution to Example 5.1

In the t-distribution table (Table 5.1), the number 2 falls between tα,8 values of 95% and 97.5% confidence (1.860 and 2.306, respectively). Hence, the yesterday’s production process can be assumed to be in control within 95% confidence but not within 97.5% confidence. The sample process average taken yesterday results in t = 2, and this number can be used to compare the variability m production to a normally occurring variability. The probability that t will exceed 1.860 is 0.05 (1 in 20 times will occur in this manner naturally), whereas the probability that t will be greater than 2.306 is 0.025 (1 in 40 times will occur in this manner naturally).

### Example 5.2

A manufacturing process for batteries has an average battery voltage output of 12 volts, with production assumed to be normally distributed. It has been decided that if a sample of 21 batteries taken from production has a sample average of 11 and sample standard deviation of 1.23, then production is declared out of control and the line is stopped. What is the confidence that this decision is a proper one to take?

Since the t distribution is symmetrical, the absolute value of t can be used. The calculated value of 3.726 falls between the ta,20 for a = 0.001 and a = 0.0005. The probability that t will exceed -3.552 is 0.001, and the probability that t will be greater than -3.849 is 0.0005. Thus, the decision is proper, since the significance of the sample occurring from the normal distribution is less 0.001 or 99.9% confidence.