Engineering Statistics 2

Course Reminders & Deadlines Pathfinder Before exercises on Statistics II due by 11:55 PM on Feb. 12th 3D Game Lab 1st deadline of 350 XP midnight Feb. 14th GameLab is 15% of your course grade No activity on the platform = B or less for your course grade Designing a Playground Quest Universal Design Project Memo Project Lab #2 due by 5 PM on Feb. 14th Note: GameLab Deadline will depend on the day your lab section runs. It should be the day your lab meets during the week of February 12th.

Activity Last week we talked about calculating measures of center (mean, median, mode) to describe data What are some of the issues that may arise from this type of analysis? What are we trying to accomplish? Think-Pair-Share (2-3 minutes) The purpose of this is to get students to think about error associated with taking a single point estimate (like mean, median). Ask students whether they would get the same estimate if they did the experiment again? Issues arise due to the inherent variation of the sample. Therefore, it is often more appropriate to display results in the form of intervals of where the population parameter could lie, instead of giving a single number. The most common method for this is the use of confidence intervals.

Interval Estimation Population How can we get a more accurate estimator of population? Therefore, better to report an interval of possible values – Confidence Interval Sample mean 𝑋 Population mean m 𝑋 is a point estimate of 𝜇, from sample Single value, not exactly equal to 𝜇 We have no idea how accurate it actually is…

Confidence Interval (CI) Estimation An interval estimate of the population parameter 𝜃 is: 𝜃 𝐿 <𝜃< 𝜃 𝑈 . Example: Interested in the true average GPA of all freshman engineering students at Rowan University A sample gives us an interval of 3.2 to 3.9. As the sample size increases, the sample variance becomes more accurate so the interval will be shorter. By the CLT, as long as n is reasonably large, the sample mean x-bar is approximately normal with mean mu and variance sigma-squared over n….this is what allows us to use the Z and T scores from the previous week! The thetas in the slide just represent some population parameter. Most often this is the population mean, represented by mu. The formula can be generalized though.

Confidence Interval The interval computed is called a: 100(1 – a)% Confidence Interval A 95% CI on mean is interpreted as follows: If we were to repeat this experiment a large number of times, and calculate a 95% CI for each experiment, in the long run 95% of those CI would contain the true value of 𝜃 In real life, we’ll only do the experiment once and we don’t know if our experiment is one of the 95% in which the CI contains the true parameter value or not. There is a trade-off between the confidence level and the width of the interval. If we wanted greater confidence that our interval contained the true parameter value (e.g. 99% confident), we could increase the confidence level. However, increasing the confidence level increases the width of the interval, and thus provides less information about the true parameter in some sense. If we follow this argument to its illogical extreme, a 100% CI for mu covers the interval from negative infinity to positive infinity. Now we are fully confident that our interval contains mu, but at the cost of having no information whatsoever about the actual value of mu.

Raritan Water System Demands 217,000 customers ADD 138 MGD (2016) MDD 190 MGD (2016) Source of Supply/Production Two WTPs Raritan Millstone (RMWTP) Canal Road (CRWTP) 74 permitted wells, 61 active Distribution System 28 gradients 3,200 miles of main 51 booster stations 41 active tanks

Supply and Demand

Normally Distributed Population Estimating the Mean Scenario 1: Estimate m Normally Distributed Population s2 unknown m unknown Sample Point Estimate of Population Mean: 𝑋 Interval Estimate of Population Mean: Central Limit Theorem…….

Estimating the Mean Determine the Confidence Interval Step 1: Set up probability statement to represent (1 – a)% of t-values Step 2: t = 𝑋 −𝜇 𝑠/ 𝑛 Step 3: Re-arrange the equation so that it is a confidence interval for the population mean, m

𝑥 −𝑡 𝛼 2 𝑠 𝑛 <𝜇< 𝑥 +𝑡 𝛼 2 𝑠 𝑛 Estimating the Mean Confidence Interval on 𝜇, with 𝜎 2 Unknown If 𝑥 and s are the mean and standard deviation of a random sample from a normal population with unknown variance 𝜎 2 , a 100 1−𝛼 % confidence interval for 𝜇 is given by 𝑥 −𝑡 𝛼 2 𝑠 𝑛 <𝜇< 𝑥 +𝑡 𝛼 2 𝑠 𝑛 where 𝑡 𝛼 2 is the t-value with v = n – 1 degrees of freedom, leaving an area of 𝛼 2 to the right.

Estimating the Mean Different samples will give different confidence intervals

Estimating the Mean – Example Example: A very costly experiment has been conducted to evaluate a new process for producing synthetic diamonds. Six diamonds have been generated by the new process with the following recorded weights (in karats) 0.46 0.61 0.52 0.48 0.57 0.54 Assume weight is normally distributed and calculate a 95% CI for the mean weight of diamonds generated by this process. Have students work on this individually for a few minutes, and then have them compare answers to those in their groups. Call on people to explain how to calculate the answer (write on the board).

Correlation Sometimes, we are curious if the 2 variables are related or not… Sample Correlation Coefficient, r A single measure of how strongly related two variables (x and y) are. Properties of r: The value of r does not depend on which of the two variables under study is labeled x and which is labeled y. The value of r is independent of the units in which x and y are measured. −1≤𝑟≤1 The square of the sample correlation coefficient (r) gives the value of R-squared in regression Correlation was mentioned in the previous anthropometry activity, but an excel function was given to calculate. We can briefly explain what correlation is.

Correlation Coefficient r near -1 r near 1 r near 0, no relationship r near 0, nonlinear relationship

Information Processing This introduces some of the principles of Information Processing that will help students understand the activity. Ask the class how they would define information? (Think, Pair, Share) Then ask the students why information processing is important in engineering design (also can be a TPS, but it doesn’t have to be)

Information Defined The reduction of uncertainty After the occurrence of an event if you are more sure of the state of the world than prior to the event, then that event has conveyed information Amount of information conveyed by an event effected by The number of events The probability of events Sequential constraints or context

H = information conveyed (bits) N = the number of possible events

Information The average information (Hav) conveyed by a series of events 1 through N Each event has a different probability, pi N Hav =  pi [log 2 (1 / pi)] i =1 The less likely the event (i.e. less probable) the more information it conveys The maximum amount of information is conveyed when the all alternative events are equally likely Log base 2 can be calculated as follows: ln(1/pi)/ln(2)

Four events each with a known probability pi D Pi 0.5 0.25 0.125 0.125 1 / Pi Log2(1/Pi) Have class work on this example 1/pi: 2,4, 8, 8 Log2(1/pi) = ln(1/pi)/ln(2): 1, 2, 3, 3 pi*(Log2(1/pi)): 0.5, 0.5, 0.375, 0.375 H av = 0.4375 Pi[Log2(1/Pi)] Hav=

Four events each with the same probability pi D Pi 0.25 0.25 0.25 0.25 1 / Pi Log2(1/Pi) 1/pi = 4 Log2(1/pi) = 2 Pi*2 = 0.5 H av = 0.5 Pi[Log2(1/Pi)] Hav=

Redundancy of Information Given N events, the amount of information conveyed can be decreased by: Unequal probabilities across events Increases in sequential constraints This decrease can be quantified as redundancy Redundancy = [1 - (Hav / Hmax)] x 100

Hick-Hyman Law Reaction time often used as a measure of cognitive processes in response to various stimuli Choice reactions occur when there is more than one potential stimuli As the number of choices (or stimuli) increases, additional cognitive processing is required  increase reaction time

Implications of the Hick-Hyman Law Reaction time increases by a constant amount each time the number of possible events N is doubled Each time the information in the stimulus is increased by one bit An increase in redundancy decreases choice reaction time Reaction times to low probability stimulus is slower than high probability stimulus

Design Recommendations Based on the Hick-Hyman Law If minimum choice reaction time is a critical task characteristic within a particular system function then: Reduce the number of possible events N Design the system so that probability of different events are different Incorporate sequential expectancy between events Reduce the total number of low probability events *These are the answers to question 7 of the information processing activity. Sequential expectancy – the extent to which people consciously assign subjective probabilities to events

Universal Design Lab #3 Objective: To begin the final steps of the redesign of your hand tool – the toy – by addressing various issues and combining your project labs into a final report. (Refer to the Project Description document on for the format of the final report). Memo: Include a set of drawings of the new tool Rationale of the toy and trade offs Important features that need to be changed or improved and why Some improvements that were left out because of cost, human factor issues were left out

Sample Memo