Activity 17.1

Coins, Presidents, and Justices

10 points

Due at the beginning of class, Friday, April 17, 2009

In the first part of this activity, you will generate some data that should have an approximately normal (or bell-shaped) distribution. In the second part, you will use the definition of standard deviation and compare the standard deviations for two different data sets.

1. Work with a partner to generate the following data.
1. a. Toss 10 coins and record the number of heads you obtained.

1. b. Repeat this 24 more times until you have a list of 25 numbers, each of them between 0 and 10.  (There will be directions in Activity 18.1 regarding how to do this in Excel.)

1. c. Retrieve the file EA17.1 Coins and Presidents.xls, and you will find the results of 35 tosses of 10 coins that someone else carried out. When you first retrieve the file, column B contains the number of times 0 heads was obtained in the 35 tosses of 10 coins, the number of times 1 head was obtained in the 35 tosses, and so on, up to the number of times 10 heads was obtained. Add your results to the list so you have a total of 60 in column B.

1. d. Create a scatterplot of these data, using one of the versions of the scatterplot with the dots connected. Be sure to add axis titles to your scatterplot. Describe what your curve looks like, including where it is "centered" and what its "spread" is. Copy and paste this scatterplot into your Word document.

1. e. Change your graph to a bar graph (instructions follow). Copy and paste this bar graph into your Word document.

1. f. If you were to draw, by-hand, a bell-shaped curve that "fits" the data in the bar graph, what would it look like? How would it compare with the curve you described in part d of this question?

2. In the second part of this activity, you will examine one measure of the spread of a data set, the standard deviation. The standard deviation plays an important role in understanding the spread of a distribution, especially a bell-shaped or normal distribution.

2. a. You'll use the data set of ages of U.S. presidents at their inauguration, which can be found on sheet 2 of the file EA17.1 Coins and Presidents.xls, and calculate the standard deviation of this data set. To do this, first find the mean (average) of the ages and store that value in cell B46. Enter the mean age in your Word document. (Source: The World Almanac and Book of Facts 2004, page 563.)

2. b. In cell C1, enter the label Deviation from the mean. In cell C2, enter an appropriate formula to subtract the mean age from the value in cell B2 and that will allow you to drag down to compute each data value minus the mean. (Remember to use $B$46 to keep the value of the mean fixed when you drag.) Drag down to compute each data value minus the mean. How many of the values in this column are negative? What property makes these values negative?

2. c. Add the values in column C and record their sum in your Word document. Type this number without using scientific notation. This number should be 0 or very, very close to 0. Is it?

2. d. Now you want to compute the square of the deviation from the mean of each data value. In cell D1, enter the label Squared deviation and use the instructions that follow to enter the squares of the deviation values in column D. Then add all the values in column D, store that number in cell D46, and enter the sum in your Word document.

2. e. Next you need to divide the sum of the squared deviations (the value in cell D46) by one less than the number of data points. There are 43 data points, so divide the value in cell D46 by 42; store this number in cell E46.

2. f. Finally, compute the standard deviation by taking the square root of the number in cell E46 (see the following instructions for a reminder of how to compute the non-negative square root of a number). Store this value in cell F46 and enter this value in your Word document.

2. g. Check the computations by using the following Excel command to compute the standard deviation of the ages in column B. Enter the value obtained in cell F47, and enter this value in your Word document.

2. h. The standard deviation of a set of data is a measure of the spread of the data values. It incorporates the sum of the squared deviations from the mean, of all data values. Suppose Bill Clinton had been 96 years old instead of 46 at his inauguration. What would change in the computations you just performed?

2. i. Change Clinton's age on your spreadsheet from 46 to 91. Notice that all of your computations change automatically. Record the new mean and the new standard deviation in your Word document.

2. j. Change Clinton's age back to its correct value of 46.

3. Go to sheet 3 of the file EA17.1 Coins and Presidents.xls, where you will find another data set. This data set contains the names and ages at time of appointment as chief justice for all the chief justices of the U.S. Supreme Court. (Source: The World Almanac and Book of Facts 2006, page 53.)

3. a. Compute the mean and standard deviation (you may just use the Excel formulas -- you do NOT need to do each step) of these ages, and Enter these values in your Word document.

3. b. Describe how the means and standard deviations of the two data sets, "Presidents' Ages" and "Supreme Court Chief Justices' Ages," compare.

3. c. Pick the maximum data value in the "Presidents' Ages" data set. Call it x, and compute its z-score by computing:

using the mean and the standard deviation of the presidents' ages. Enter this z-score in your Word document.

3. d. Find the z-score for the largest data value in the "Supreme Court Chief Justices' Ages" data set, using the mean and standard deviation of the chief justices' ages. Enter this z-score in your Word document.

3. e. What do the two z-scores tell you?

Summary
In the first part of this activity, you generated data and created a graph to see that the data have an approximately normal distribution. In the second part of the activity, you worked with the standard deviation to see the impact of a large data value on this measure of spread. You also compared data from two data sets by looking at z-scores.