# UTPA STEM/CBI Courses/Measurements and Instrumentation/Probability & Statistics

Course Title: Measurements & Instrumentation

Lecture Topic: Probability & Statistics

Instructor: Dr. Isaac Choutapalli

Institution: UTPA

## Backwards Design[edit]

**Course Objectives**

**Primary Objectives**- By the next class period students will be able to:- Learn about Mean
- Learn about Standard deviation
- Learn about Variance
- Learn about the Difference between mean and sample mean
- Learn about the Different kinds of probability distributions
- learn about Precision interval
- Learn about Number of samples required for good statistical estimation
- Learn error estimation of data samples
- Learn about confidence intervals
- Learn about data outlier detection
- Regression Analysis

**Sub Objectives**- The objectives will require that students be able to:- Learn about different kinds of probability distributions

**Difficulties**- Students may have difficulty:- Performing integrations of mathematical expressions

**Real-World Contexts**- There are many ways that students can use this material in the real-world, such as:- Reliability of measured data by estimating the random error
- Eliminating data outliers in real-world measurements
- Estimating the required number of measurements, i.e. gathering a good data sample

**Model of Knowledge**

**Concept Map**- Mean and standard deviation of a sample
- Probability of the occurrence of an event
- Different types of probability distribution curves
- Integration of probability distribution functions

**Content Priorities****Enduring Understanding**- Mean, standard deviation and variance
- Probability distribution functions

**Important to Do and Know**- Integration and differentiation
- Simple programming using MATLAB

**Worth Being Familiar with**- Flow, temperature and pressure measurement
- Measurement instrument specifications

**Assessment of Learning**

**Formative Assessment**- In Class (groups)
- Calculate statistical quantities from a given sample and obtain the precision intervals based on the calculated statistical quantities.

- Homework (individual)
- Read the chapter before coming to the class.
- Quiz after the lecture is delivered

- In Class (groups)
**Summative Assessment**- Quiz/mid-term exam
- Laboratory experiment

## Legacy Cycle[edit]

**OBJECTIVE**

By the next class period, students will be able to:

- Learn about Mean
- Learn about Standard deviation
- Learn about Variance
- Learn about the Difference between mean and sample mean
- Learn about the Different kinds of probability distributions
- learn about Precision interval
- Learn about Number of samples required for good statistical estimation
- Learn error estimation of data samples
- Learn about confidence intervals
- Learn about data outlier detection
- Regression Analysis

**THE CHALLENGE**

You have to decide whether or not the concrete being used to pour a structural footing is good enough. If the concrete is weak, the stability of the entire structure will be at stake and the lives of the people living inside this building will be at stake. You have two sets of samples with average strengths of 2800 lb/in^{2} and 3100 lb/in^{2}. Which would you choose to be poured for the structural footing? The state codes require that the strength be at least 3000 lb/in^{2}.

**GENERATE IDEAS**

- What does the average of the samples mean?
- Do you think the number of samples used in the calculation of average play an important role?
- If some of them fall below 3000 lb/in
^{2}, what would your suggestion be? Would you repour the entire footing? - If some of them fall below 3000 ln/in
^{2}and most of them above 3000 lb/in^{2}, would you repour the footing? - On what basis would you decide whether or not you should repour the footing?
- How many samples would you test to make sure that you don't put the lives of people at stake?

**MULTIPLE PERSPECTIVES**

- Instructor will get the students' view points first
- The instructor will also provide his view point and compare it those of the students'.

**RESEARCH & REVISE**

- Mean strength of the samples.
- Mean strength in light of the standard deviation of the samples.
- What are the lowest and highest strengths that you obtain from testing one given set of samples?
- Think about if that is a good indication of the strength of the concrete that would be used in the footing.
- Now, for example, if you test another set of samples and you find that the new mean is different from that obtained the firt time, how would you relate the two.
- This sets the stage for the introduction to the concept of standard deviation of means.
- Concept of students t-distribution.
- Concept of different probability distributions such as the gaussian, poisson, binomial and log-normal distributions.
- Determination of the probability of occurence based on the students t-distribution and the standard deviation of the sample mean.
- Examination of the confidence/precision interval or the error.
- In light of this precision interval, re-examine the highest and lowest strengths of the concrete from the samples tested.
- Effect of the number of samples of the mean and standard deviation.

**TEST YOUR METTLE**

Students will receive feedback and their understanding will be tested by presenting them with a quiz.

**GO PUBLIC**

The students will now be able to answer the questions posed at the beginning of the challenge cycle and will be asked to provide a brief explanation for each question.

## Pre-Lesson Quiz[edit]

- How is the behavior of a random variable defined?
- What do you understand by the term "Statistical Distributions"?
- What are confidence intervals in a Gaussian distribution?
- What are the sample statistical parameters of a sample population?
- What is the criteria for statistical rejection of outliers from a sample?
- What is the criteria for the minimum number of measurements required for an experiment?

## Test Your Mettle Quiz[edit]

- The setup for grinding a type of bearing is considered under control if the bearings have a mean diameter of 5.0mm. Normal procedure is to measure 30 bearings each 10 minutes to monitor production. Production rates are 1000 per minute. What action do you recommend if such a trial sampling of 30 bearings shows a mean of 5.060mm with a standard deviation of 0.0025mm?
- Based on 51 measurements of a time-dependent electrical signal, the standard deviation is 1.52V. Find the 95% confidence interval in its true mean value. How many more measurements would be required to provide a 95% confidence interval in the true mean to within (+/-)0.28V?
- A conductor is insulated using an enameling process. It is known that the number of insulation breaks per meter of wire is 0.07. What is the probability of finding
*x*breaks in a piece of wire 5m long? Use the Poisson distribution. - An optical velocity measuring instrument provides an updated signal on the passage of small particles through its focal point. Suppose the average number of particles passing in a given time interval is 4. Estimate the probability of observing
*x*particle passages in a given time.