Authentic Learning Task (ALT)#3: Return to Pennyville

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Overview

This Authentic Learning Task (ALT) focuses on the process of taking the mean from a sample and using it to estimate the average or mean of a population. The Central Limit Theorem is introduced and helps control of the accuracy of our estimate. You begin where you left off in ALT #2, using Pennyville again as a population to illustrate these concepts.

After completeing this ALT, you should be able to demonstrate the following competencies:

• Using sample data, make predictions based on the foundational principles of the Central Limit Theorem (Comp. 3).

Materials and Equipment
No materials or equipment are needed to complete the ALT.

Safety and Disposal
No special safety or disposal procedures are required.

Pre-Activity

Empirical Rule, Standard Error of the Mean, and Confidence Intervals for the Mean

In this ALT, you follow-up and use some of the results from your group’s penny data collected in the last activity.

As you observed in the last activity, the sample averages tend to be normally distributed even though the underlying distribution is skewed or uniform. This principal is foundational for the effectiveness of Average Control Charts for variables. It is not a requirement for the underlying distribution to be normally distributed because of the central limit theory. The distribution of averages (means) will be normally distributed.

In this exercise, you will focus on the empirical rule and apply the central limit theorem concept in the application of the standard error of the mean statistic and corresponding confidence intervals of the mean.

Normal Distribution and the Empirical Rule

The empirical rule states that, given the standard deviation and the mean values for any normal distribution, we can predict, using the standard normal table, the probability of occurence of any specific value.

Complete the following learning objects on the Normal Distribution and the reference materials on the Normal distribution.

Complete the following learning objects on:

• The Empirical Rule
• The Standard Normal Distribution
  
  Reference materials about the Normal Distribution can be found here.
  
  The Vassar College site contains references and statistical calculators for your use.
  

Confidence Interval for the Mean

Complete the “Confidence Interval for the Mean” learning object by downloading the Normal Distribution Excel Spreadsheet or use a Normal distribution table and calculator.

The confidence interval simulator will provide insight into the relationship between the probability and the sample.

A very complete online reference source including statistical calculators is found at Vassar College

Self-Assessment

Work the following problem:
Five ball bearings were randomly sampled from a production line and their diameters were measured. The results in millimeters:

10.253,10.255.10.251, 10.251. 10.254

• Calculate the sample mean and sample standard deviation.
• Calculate the standard error of the mean.
• Determine the 90% confidence interval on the mean.

What should the minimum specification for the diameter be in order to assure that only 1/100,000 would fall outside of the lower tolerance?

Click here for the worked answers to problem.

Activity

1. Your facilitator will place you and the other participants in teams of two.
2. In your team, take 30 pennies from Pennyville.
3. Calculate and record the average and standard deviation from your sample of 30 pennies.
4. Construct the interval ( ) under the sampling distribution on the flipchart. Use the average of the sample standard deviations from ALT #2 for to calculate.
5. Draw a vertical line centered at the real average of the population down through the sampling distribution and all the intervals.
6. Complete:
   • Data Sheet: Interpretation
   • Data Sheet: Central Limit Theorem
   • Data Sheet: No More Pennies
   

Post-Activity

Your team should now post all of its results from the activity to the Discussion Board.

Assignment
There are no instructions to prepare for ALT #4: How High?

Assessment
Your facilitator may use Assessment Sheet: Return to Pennyville to evaluate your results from the activity and your posting to the Discussion Board.

Go to next ALT

Probability, Sampling, and Hypothesis Testing Home Page

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