Unit 8: Normal Distribution

Read chapters 18 and 19 of FPP and do the following review exercises:
Chapter 17 (pages 304-306): 7, 8, 9, 12
Chapter 18 (pages 327-329): 2, 4, 5, 6, 7, 12, 13, 14
Chapter 19 (pages 351-353): 5, 10, 11

Just Say No--To Bad Science," Newsweek, "On Science" column, May 7, 2007, page 57.
Available on LexisNexis.

Possible Essay Questions:

• The column finds many problems with various studies of the same topic. How would you design a study of that topic?
• It is sometimes claimed that no educational experiment ever fails, because the experimenter is an enthusiastic teacher who got the idea. Does this relate to the present column, and if so, how?

Computer project:

Description

How close should the count of heads on 40 coin tosses be to 20? What is the spread in this count? Notice that the spread of the count is a different thing than the spread of the collection of flip results. The spread of the count is called the standard error(SE). The spread in the collection of flips is called the standard deviation (SD). Notice also that this means that the SE is the SD of many attempts to count heads.

In this project we explore how the spread of the counts and the average of the counts change as the number of flips increase.

Preliminary Write-up

Using techniques from class and before going to the computer write up what theory predicts for the results as follows:
1. Estimate the number of heads you would expect to get and the spread in this number of heads for three cases of the number of flips: 10 flips, 40 flips and 160 flips.
2. As the number of flips is increased by a factor of 4 from 10 to 40 and from 40 to 160, how does this affect the expected value of the count (what factor of increase occurs)? By what factor does the standard error increase?
3. Consider doing the previous experiment 30 times and trying to get exactly half the flips being heads as many times out of the 30 as possible. Would you get exactly half more often with a 10 flip experiment, 40 flip experiment or 160 flip experiment?

Simulations

Use Excel to perform the following coin tossing simulations. Use the random number (RAND) and IF functions to simulate a coin toss. (See the simulation instructions if needed.) Structure your spreadsheet in an organized fashion with, for example, each simulation being one column. The result of the simulation can then be placed conveniently at the top or bottom of the column. After you check to make sure one simulation is working you can copy that column 30 times and create summary statistics for the entire 30 simulations somewhere convenient.
1. Simulate tossing a fair coin 10 times and counting the number of heads. Do the simulation 30 times and compute the average and SD of the 30 counts. Note that the SD function is used here to measure the SE of the counts. Theory says the average of these 30 counts should be close to the expected value and the spread of the counts should be close to the SE of the counts.
2. Do 30 simulations of tossing a fair coin 40 times and counting the number of heads. Compute the average and SD of the 30 counts.
3. Do 30 simulations of tossing a fair coin 160 times and counting the number of heads. Compute the average and SD of the 30 counts.
4. The numbers of tosses was increased by a factor of 4 from (a) to (b) and from (b) to (c). How did this affect the average and SD (in terms of factor of increase -- i.e., unchanged, factor of 4, factor of 0.5, etc.)? Compare these factor increases with your preliminary predictions from 2) above.
5. Looking at the counts for each set of simulations, how many times out of 30 did you get exactly half of the flips being heads? In which case, (a), (b) or (c), are you most likely to get heads on exactly half the number of tosses? Compare with your preliminary theory from 3) above.
The text passage provides a detailed description of a computer project involving coin toss simulations, including theoretical predictions and practical simulations using Excel. It also includes a set of preliminary write-up questions and simulation tasks.
multiple_questions

the passage contains multiple questions and tasks that require mathematical and statistical analysis, such as estimating the number of heads in coin tosses, calculating the standard error and standard deviation, and comparing theoretical predictions with simulation results.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Unit 8: Normal Distribution

Read chapters 18 and 19 of FPP and do the following review exercises:
Chapter 17 (pages 304-306): 7, 8, 9, 12
Chapter 18 (pages 327-329): 2, 4, 5, 6, 7, 12, 13, 14
Chapter 19 (pages 351-353): 5, 10, 11

Just Say No--To Bad Science," Newsweek, "On Science" column, May 7, 2007, page 57.
Available on LexisNexis.

Possible Essay Questions:

• The column finds many problems with various studies of the same topic. How would you design a study of that topic?
• It is sometimes claimed that no educational experiment ever fails, because the experimenter is an enthusiastic teacher who got the idea. Does this relate to the present column, and if so, how?

Computer project:

Description

How close should the count of heads on 40 coin tosses be to 20? What is the spread in this count? Notice that the spread of the count is a different thing than the spread of the collection of flip results. The spread of the count is called the standard error(SE). The spread in the collection of flips is called the standard deviation (SD). Notice also that this means that the SE is the SD of many attempts to count heads.

In this project we explore how the spread of the counts and the average of the counts change as the number of flips increase.

Preliminary Write-up

Using techniques from class and before going to the computer write up what theory predicts for the results as follows:
1. Estimate the number of heads you would expect to get and the spread in this number of heads for three cases of the number of flips: 10 flips, 40 flips and 160 flips.
2. As the number of flips is increased by a factor of 4 from 10 to 40 and from 40 to 160, how does this affect the expected value of the count (what factor of increase occurs)? By what factor does the standard error increase?
3. Consider doing the previous experiment 30 times and trying to get exactly half the flips being heads as many times out of the 30 as possible. Would you get exactly half more often with a 10 flip experiment, 40 flip experiment or 160 flip experiment?

Simulations

Use Excel to perform the following coin tossing simulations. Use the random number (RAND) and IF functions to simulate a coin toss. (See the simulation instructions if needed.) Structure your spreadsheet in an organized fashion with, for example, each simulation being one column. The result of the simulation can then be placed conveniently at the top or bottom of the column. After you check to make sure one simulation is working you can copy that column 30 times and create summary statistics for the entire 30 simulations somewhere convenient.
1. Simulate tossing a fair coin 10 times and counting the number of heads. Do the simulation 30 times and compute the average and SD of the 30 counts. Note that the SD function is used here to measure the SE of the counts. Theory says the average of these 30 counts should be close to the expected value and the spread of the counts should be close to the SE of the counts.
2. Do 30 simulations of tossing a fair coin 40 times and counting the number of heads. Compute the average and SD of the 30 counts.
3. Do 30 simulations of tossing a fair coin 160 times and counting the number of heads. Compute the average and SD of the 30 counts.
4. The numbers of tosses was increased by a factor of 4 from (a) to (b) and from (b) to (c). How did this affect the average and SD (in terms of factor of increase -- i.e., unchanged, factor of 4, factor of 0.5, etc.)? Compare these factor increases with your preliminary predictions from 2) above.
5. Looking at the counts for each set of simulations, how many times out of 30 did you get exactly half of the flips being heads? In which case, (a), (b) or (c), are you most likely to get heads on exactly half the number of tosses? Compare with your preliminary theory from 3) above.
user
The text passage provides a detailed description of a computer project involving coin toss simulations, including theoretical predictions and practical simulations using Excel. It also includes a set of preliminary write-up questions and simulation tasks.

TYPE: MULTIPLE_QUESTIONS

The passage contains multiple questions and tasks that require mathematical and statistical analysis, such as estimating the number of heads in coin tosses, calculating the standard error and standard deviation, and comparing theoretical predictions with simulation results.