# Unit 8: The Normal Distribution

Read chapters 18 and 19 of FPP and do the following review exercises:
Chapter 18 (pages 327-329): 2, 4, 5, 6, 7, 12, 13, 14
Chapter 19 (pages 351-353): 5, 10, 11

"Texas death row inmate loses Supreme Court appeal to be spared because of low IQ", CBS News website, August 7, 2012.

## Possible essay questions:

• Population IQ scores are very close to normally distributed, with standard deviation about 15 points. About what fraction of people fail to meet the "minimum competency standard" to stand trial? About what fraction of people have lower IQ's than the defendant in this case?
• Does the rule of ineligibility for the death penaly based on IQ make sense? Why or why not?

## Computer project:

1. Use a spreadsheet to carry out 30 simulations of the two strategies (i) and (ii) discussed in Problem 9 of Chapter 17 (page 305). How do your simulation results compare with your theoretical answers for 9a), b) and c)? Explain.
2. A multiple-choice quiz has 25 questions. Each question has 5 possible answers, only one of which is correct. Four points are given for each correct answer, but a point is taken off for a wrong answer. Use a spreadsheet to simulate the scores of 100 students answering all of the questions at random. How many, and what percentage, of the students score at least 30 points? Compare your results with those obtained by theoretical computation using expected value and standard error. Repeat the simulation and comparison with 400 simulated students guessing randomly.

#### Simulation Tips:

Use each column of the spreadsheet as a single simulation of 1000 plays. The number in each cell should contain the winnings on one spin of the roulette wheel. [i.e. for 9(i) -- the columns strategy -- the contents of a cell might be =IF(RAND()<(12/38),2,-1).] Below each simulation, calculate the total winnings.

To count how many of the simulations "came out ahead", won 100 or lost 100, use the FREQUENCY function with the "data array" being the row of total winnings, and the "bin array" being -100,0,100 (put these numbers in three adjoining cells.) The function FREQUENCY counts how many times an outcome was less than each bin label: So, the number next to -100 is the number of times we lost more than \$100. The number next to 0 is the count between -\$100 and \$0. The number next to \$100 is the count between \$0 and \$100 and the last number is the count greater than \$100. (Note: four counts for three bin numbers.)

To save paper, print out only the bins and frequencies.

A source of help for this project is a video outlining a similar project.

Revised: 27 August 2012. Questions to: dlantz@colgate.edu