TENNESSEE STATE UNIVERSITY

COLLEGE OF ENGINEERING

DEPARTMENT OF APPLIED AND INDUSTRIAL TECHNOLOGIES

AITT 3480 STATISICAL QUALITY CONTROL Project 5

Statistical Methods Useful in Quality Control and Improvement

Due: Thursday – November 5, 2020 Objective:

§ Investigate histograms, binomial distribution and linear regression analysis Introduction:

Statistics is a collection of techniques useful for making decisions about a process or population based on an analysis of the information contained in a sample from that population. Statistical methods play a vital role in quality control and improvement. They provide the principal means by which a product is sampled, tested, and evaluated, and the information in those data is used to control and improve the process and the product. Furthermore, statistics is the language in which development engineers, manufacturing, procurement, management, and other functional components of the business communicate about quality. In this project we investigate the following statistical methods: histograms, binomial distribution and linear regression.

AITT 3480 STATISICAL QUALITY CONTROL Project 5

Histograms

A histogram is an approximate representation of the distribution of numerical data. To construct a histogram, the first step is to “bin” (or “bucket”) the range of values—that is, divide the entire range of values into a series of intervals—and then count how many values fall into each interval. The bins are usually specified as consecutive, non-overlapping intervals of a variable. The bins (intervals) must be adjacent and are often (but not required to be) of equal size.

If the bins are of equal size, a rectangle is erected over the bin with height proportional to the frequency—the number of cases in each bin. A histogram may also be normalized to display “relative” frequencies. It then shows the proportion of cases that fall into each of several categories, with the sum of the heights equaling 1.

The histogram is one of the seven basic tools of quality control. Histograms are sometimes confused with bar charts. A histogram is used for continuous data, where the bins represent ranges of data, while a bar chart is a plot of categorical variables.

Binomial Distribution

A binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times. The binomial is a type of distribution that has two possible outcomes (the prefix “bi” means two, or twice). For example, a coin toss has only two possible outcomes: heads or tails and taking a test could have two possible outcomes: pass or fail.

Figure 1. A Binomial Distribution shows either (S)uccess or (F)ailure.

§ The first variable in the binomial formula, n, stands for the number of times the experiment runs.

§ The second variable, p, represents the probability of one specific outcome.

For example, let’s suppose you wanted to know the probability of getting a 1 on a die roll. if you were to roll a die 20 times, the probability of rolling a one on any throw is 1/6. Roll twenty times and you have a binomial distribution of (n=20, p=1/6). SUCCESS would be “roll a one” and FAILURE would be “roll anything else.” If the outcome in question was the probability of the die landing on an even number, the binomial distribution would then become (n=20, p=1/2). That’s because your probability of throwing an even number is one half.

Linear Regression

AITT 3480 STATISICAL QUALITY CONTROL Project 5

Linear regression is the most widely used statistical technique; it is a way to model a relationship between two sets of variables. The result is a linear regression equation that can be used to make predictions about data.

In linear regression the X variable is sometimes called the independent variable and the Y variable is called the dependent variable. Simple linear regression plots one independent variable X against one dependent variable Y. Technically, in regression analysis, the independent variable is usually called the predictor variable and the dependent variable is called the criterion variable. However, many people just call them the independent and dependent variables. More advanced regression techniques (like multiple regression) use multiple independent variables.

Regression analysis can result in linear or nonlinear graphs. A linear regression is where the relationships between your variables can be described with a straight line. Non-linear regressions produce curved lines.

Figure 2. Simple linear regression for the amount of rainfall per year.

Assignment:

1. The data shown in Table 1 are chemical process yield readings on successive days (read down, then across). (Must show how calculated)

(a) Calculate the sample average and standard deviation (b) Construct a histogram

Table 1

- Binomial distribution: Given that a coin has two sides (heads and tails). Determine the probability distribution of the number of Heads from flipping the coin 7 times. (Must show how calculated)
- Linear Regression: Perform the following statistical analysis on the data set in Table 2:

AITT 3480 STATISICAL QUALITY CONTROL Project 5

Reporting:

- a) Regression using least square method
- b) Regression using R2 method

Table 2

X |
Y |

2 |
20 |

10 |
40 |

6 |
10 |

8 |
30 |

2 |
40 |

12 |
25 |

6 |
35 |

4 |
30 |

8 |
50 |

10 |
15 |

4 |
45 |

X = number of category 2 hurricanes that made landfall per year Y = number of persons per 100,000 without power

This project reporting requires that the following information be completed for submission. The report must be word processed with size 12-font. The report should have at least the following format:

I.

II. III. IV . V .

AITT 3480 STATISICAL QUALITY CONTROL Project 5

Reference:

- Creating a Histogram
https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-data- statistics/histograms/v/histograms-intro

- How to Determine Histogram Bin Width and Bin Intervals https://www.qimacros.com/histogram-excel/how-to-determine-histogram-bin- interval/
- Binomial random variables
https://www.khanacademy.or /math/statistics-probability/random-va tats- library/binomial-random-variables/v/binomial-distribution?modal=1

- Visualizing a binomial distribution
https://www.khanacademy.org/math/statistics-probability/random-variables- statslibrary/binomial-random-variables/v/visualizing-a-binomial-distribution?modal=1

- Binomial probability example https://www.khanacademy.org/math/statistics-probability/random-variables-stats- library/binomial-random-variables/v/probability-of-making-2-shots-in-6- attempts?modal=1
- An Introduction to Linear Regression Analysis https://www.youtube.com/watch?v=zPG4NjIkCjc
- How to calculate linear regression using least square method https://www.youtube.com/watch?v=JvS2triCgOY
- How to Calculate R Squared Using Regression Analysis https://www.youtube.com/watch?v=w2FKXOa0HGA
- Standard Error of the Estimate used in Regression Analysis (Mean Square Error) https://www.youtube.com/watch?v=r-txC-dpI-E