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64 PCB007 MAGAZINE I DECEMBER 2024 e general interpretation of p-values is as follows: < 0.05 are statistically significant 0.05 ≤ p-value ≤ 0.10 may have a practical difference > 0.10 is generally considered non-significant Statistically significant means that the results in the data are not likely explainable by chance alone. Practical difference requires the pro- cess engineer to logically determine if the population mean or variance differences have any practical value. Non-significant means the results in the data lie within the limits of chance. Microsoft Excel Data Analysis ToolPak e Analysis ToolPak provides tools for sta- tistical data analysis. It offers different kinds of statistical analysis, including t-Tests and F-tests. is enables the exploration and statis- tical analysis of data without using any external soware. ToolPak is, therefore, useful to pro- cess engineers working with datasets 3 . To access ToolPak, click "Data Analysis" in the "Analysis" group on the "Data" tab. Sup- pose the Data Analysis command is not avail- able. In that case, you can load the Analysis ToolPak add-in program by following these steps: 1. Click the File tab, click Options, and then click the Add-Ins category. 2. In the Manage box, select Excel Add-ins and then click Go. Go to Tools > Excel Add-ins in the file menu if you're using Excel for Mac. 3. In the Add-Ins box, check the Analysis ToolPak check box and click OK. If Anal- ysis ToolPak is not listed in the Add-Ins available box, click Browse to locate it. If you are prompted that the Analysis Tool- Pak is not currently installed on your computer, click Yes to install it 3 . Figure 1 shows some of the analysis tools available in ToolPak. Statistics and Probability Statistics is a branch of mathematics that collects, analyzes, interprets, and presents numerical data. It is not merely the science of analyzing data but the art and science of collecting and analyzing data. Statistics are tools to help us; they do not replace the pro- cess engineer's skill and intelligence. Statistics uses hypotheses, experiments, and hypothesis testing. With hypotheses, a null (H 0 ) and an alter- native (H 1 ) are put forth. e null hypothe- sis states that two population means or vari- ances are equal. e alternative hypothesis is a statement that the two population means or variances are not equal (two-tailed). However, the alternative hypothesis can be either one or two-tailed. e one-tail tests for either inferi- ority or superiority, while the two-tail tests for parity (not equal). Typically, we test for parity, which tests for both inferiority and superiority. With hypothesis testing, we either accept the null or alternative hypothesis based on a probability value (alpha). Formally, the null and two-tailed alternative hypotheses for the mean and variance are written as follows: Mean: H 0 : µ 1 = µ 2 H 1 : µ 1 ≠ µ 2 Variance: H 0 : σ 2 1 = σ 2 2 H 1 : σ 2 1 ≠ σ 2 2 Alpha defines the test's sensitivity. A value of 0.05 implies that the null hypothesis is rejected 5% of the time when it is, in fact, true. e choice of alpha is somewhat arbitrary, although, in practice, a value of 0.05 is typi- cal. Critical values are cut-off values that define regions where the test statistic is unlikely to lie 2 . Probability is a branch of mathematics that deals with the occurrence of a random event. P-values are expressed from zero to one and are the probability of an event having occurred.