TI–MC Chapter 1 Exploring Statistics with the TI-83/84

Mr. Billings is a high school math teacher at Mooseville High School. Recently he gave a test to his two classes of 9th grade students. The day after the test was given and all of the tests had been graded, one of his students asked the question, “Which class did better, the morning or the afternoon?” His answer was, “I’m not sure. Both seemed about the same, but I should use math to check.” Mr. Billings decided to take out his graphing calculator and use its tools to compare these two classes.

The grades for the two classes are as follows:

These scores will be used for almost all the calculator activities for chapter 1.

TI-MC Activity #1 Finding the Mean

Twenty students in Mr. Billing’s morning MATH Connections class took a test. The scores on the test were as follows:

69,92,79,49,83,67,58,83,78,99,60,87,100,74,64,55,81,81,87,91

1. Using the home screen of your calculator, find the Mean (average) score for this class on the test. To go to the home screen press ` M for QUIT. If your home screen is not clear, you can press Cto erase items written there. Now enter the data on the home screen to find the mean.

Brian’s calculator window looked like this.

Steve’s calculator window looked like this.

After pressing e which of these students do you think will get the correct answer for the mean of the test scores? Why? What answer did you get?

2. Your calculator also will find the mean of these scores for you. In order to do this you must enter the test scores in a list. To enter the scores in a list press theS key. You will see this screen

Presseor1to see this screen

If your lists have numbers in them you can clear them using the following steps. Use the:key to move the cursor up to highlight the name of the list. Your screen will look like this

Now by pressing C and then e you can empty the list.

Important Note: Some students make the mistake of pressing the d key when trying to clear a list. This is a mistake that will delete the entire list, not just the contents of the list. Be certain that you do not do this.

There is also a “fast” way to clear all lists. To do this press the `+ keys for MEM (memory). You will see this screen

Choose 4: ClrAllLists. This will paste the command on your home screen. After you press e the screen will change to show you that the lists have been cleared. You will see this screen

This way of clearing all lists will clear all of the lists on the list screen (L1-L6). It will not clear “user defined lists” that have been created and named to save data. You should be careful using this method to clear a single list. Be certain that you do not want the data in the other five lists for future use.

Now enter your data into L1. You do this by typing the number and then pressing e for each number. If you make a mistake you can simply type over the incorrect number. If you leave out a number, you can insert it. INS (insert) adds the number to the list above the number that you highlight. So, to insert a number, highlight the number that comes after it, press `d to INS, type the missing number and then press e .

When all of the data has been entered in a list, you can find the mean of that list. To do this, go to the home screen then press `S for LIST. Then press the right arrow key twice to highlight MATH. Your screen should look like this

When you choose 3:mean( this will be pasted to your home screen. You will see the screen to the right.

Type in the name of the list that you want. The list names are found over the number keys. L1 is over 1 and L2 is over 2 and so on… As your data is in L1, choose L1. This screen will appear

When you press e the answer will appear. What was the mean for Mr. Billing’s morning class that you found using this method? Does the answer agree with the answer that you found using your home screen?

3. Another way to find the mean of a list is to press S> for CALC. Your screen will look like this:

When you choose 1:1-Var Stats by pressing e or 1 the command will be pasted onto the home screen. The blinking cursor is waiting for you to tell the calculator which list you want it to perform 1-Var Stats on. You can type the appropriate list in here before pressing e . The calculator defaults to L1 so if your data is in L1 you do not have to tell the calculator which list to use. This saves time and is a reason to use L1 rather than other lists if you just have one data set. When you do press e on this screen analyzed information will appear. The first item on this list is the mean of the data set. The symbol showing it is an “X” with a “bar” above it. When you find the mean this way, does it match the answers that you found previously?

4. Now that you know how to find the mean of a data set, use these scores from Mr. Billing’s afternoon class to determine the mean score for his other class on the test. The scores from the afternoon class were

71,62,70,79,82,66,65,76,76,70,62,77,71,68,82,63,76,74,84,70

Place this new data in L2 so that you will still have L1 for future use. What is the mean score for the afternoon class? Which of the two classes obtained the higher mean score on this test? Which of the classes do you think is “smarter” based on this data?

TI-MC Activity #2 Histograms

Sometimes it is easier to “see” data by placing it in another form other than numeric. One way to do this is to make a histogram from a data set. Here is how to make a histogram using the scores in Mr. Billing’s morning class.

1. If the test scores (data) are not still in L1, place them there. This can be done by retyping them or moving them there if you have stored them in a “named” list. Press `! for STAT PLOT. Your screen will look like this:

If you screen does not look like this (some plots are On), you can choose 4:Plots…Off to turn them all off. On this screen you can choose any of three plots and turn plots Off or On. You only need to choose one plot for one set of data. Choose 1:Plot 1…Off. You will then see this screen

Work your way through this screen turning the plot on by pressing e when On is highlighted and e when the appropriate type is highlighted. The third icon of the six listed is the histogram. When you press e to select histogram, you will have only an Xlist as you have only one data set. If your screen does not identify L1 for the Xlist type L1 in there using `1 or `S and choosing L1. Your screen should look like this:

When you press the % key a histogram should appear. In reality one will probably not show on the screen as the viewing “window” probably needs to be adjusted. Press @. You should see the screen to the right.

An X-window from 45 to 105 is wide enough to see all of the values. The y-window is a frequency scale so you will need to estimate about how many data values will fall into each “bar” on the histogram. Key to the window setting when plotting a histogram is the X-scl setting. This tells the calculator how wide to make each bar of the histogram. The window to the right will produce a histogram with a good viewing window.

The X-scl of 5 makes each bar five units wide. The Y-min of -2 moves the entire graph up on the screen so that you can see what might be written below the graph. Try experimenting with different values in the window setting to see how the graph changes. Changes in the X-scale are particularly interesting. If you press $ and use the and keys to trace on the histogram you will see the data associated with each “bar.”

  1. Now that you have learned how to make a histogram using the test scores from the morning class, make a histogram for the afternoon class. That data should be placed in L2 so that you do not lose the data from the first class. After you have produced both of these histograms make a sketch of each.

Morning Class Afternoon Class

3. Do you think that these histograms help you to compare the two classes? Now, which class do you think is the “smartest?” Why? Where is the mean located on each histogram? Is this a surprise to you?

TI–MC Activity #3 5-Number Summary

Recall the two sets of grade data from Mr. Billings’ class that appeared in earlier activities. He gave a test to two different classes, and he is trying to figure out if one class did significantly better than the other. The morning class is represented by Ω and the afternoon class by æ.

In this worksheet, you will be comparing data with a new type of representation called a 5-number summary, which can easily be found by sorting the data.

1. Unless using an alternative method, repeat the instructions from Workbook 1 to put Data Set A intoΩand Data Set B intoæ. If they are already in your calculator you can view them by pressing Se.

2. Press `M(QUIT) to go back to the home screen. Press Se to proceed to the STAT menu and choose “2:sortA(" to sort the data into ascending order. Press `1 (Ω) to indicate that you are sorting Ω, then close the parentheses and press e. You have just sorted Ω.

3. Now press Se to examine your lists because your five-number summaries can be found within. The first and last are easiest to find.

a. What is the minimum for Ω?

b. What is the maximum for Ω?

3. (continued)

To find the other three we will have to examine the quartiles, which are numbers that divide the data into four equal sections. Directly in the middle, you will find the median, a number that splits the second and third quartiles down the middle. Scroll down to find the middle two numbers in the list. aBecause we have an even number of data points, there is no one point that represents the middle. The median will be the average of these two middle numbers.

c. What is the medianfor Ω?

The 1st quartile can be found by calculating the median of the bottom half of your data.

c. What number represents the 1st quartilefor Ω?

d. How did you get it?

The 3rd quartile can be found be calculating the median for the top half of your data.

c. What number represents the 3rd quartilefor Ω?

d. How did you get it?

  1. You now have a five number summary. Write these five numbers in order from lowest to highest.

______

Minimum 1stQ. Median 3rd Q. Maximum

4. Repeat this process for æ.

______

Minimum 1st Q. Median 3rd Q. Maximum

5.Of course your calculator can calculate this quite easily. Go back to your main screen and press S . Then go over to CALC and choose “1:1-Var Stats.” After choosing and returning to your main screen, press `1 to choose Ω as the list your will be evaluating. Press e to execute.

Although you will see many summary numbers, the 5-number summary is not visible on this screen. Note the down arrow that appears in the lower left of the screen. That indicates there is more to be seen. Press ; (down arrow) multiple times to view the 5-number summary.

  1. Write the 5-number summary in the space below.

______

Minimum 1st Q. Median 3rd Q. Maximum

  1. Do they match your previous answers? If they don’t explain what you may have done incorrectly.

6.a. Repeat problem #5 with æ. Write your results below.

______

Minimum 1st Q. Median 3rd Q. Maximum

  1. Do they match your previous answers? If they don’t explain what you may have done incorrectly.

7. The power of the 5-number summary is the ability to compare different sets of data. Mr. Billings wants to use the 5-number summary to see if one class’s scores are better than the other’s. Which class do you think did better, the morning (Ω) or the afternoon (æ)? Use the 5-number as evidence to help explain your choice.

TI–MC Activity #4 Examining Boxplots

Recall the two sets of grade data from Mr. Billings’ class that appeared in earlier activities. He gave a test to two different classes, and he is trying to figure out if one class did significantly better than the other. The morning class is represented by Ω and the afternoon class by æ.

You will further compare these two sets of data by building on the previous activity, 5-Number Summaries. Recall that a 5-number summary displays the following: Minimum, 1st Quartile, Median, 2nd Quartile, Maximum. Using these, one can discover if there are significant differences in data. A graphical version of the 5-number summary is known as a Boxplot.

1. Unless using an alternative method, repeat the instructions from Activity 1 to put Data Set A intoΩand Data Set B intoæ. If they are already in your calculator you can view them by pressing Se.

Note: The data in the screen shown have been sorted.

2. Press ! to check that there are no selected equations in the ! screen. If there are, you can press C to get rid of each one.

3. Press`! to go to the STAT PLOTS screen. Choose 4:PlotsOff to clear all existing plots. Press e to execute this command.

  1. Press`! again to return to the STAT PLOTS screen. You are now ready to begin creating your Boxplot. Choose “1:Plot1" by pressing e. Turn on the plot by pressing e. Press ; to go down to Type. Choose the 5th icon (∆) by pressing four times. For the Xlist, choose Ω by pressing `1. Your Boxplot is now ready for graphing
  1. Press the @ button and enter in the values shown to the right. Press % to view your results.
  1. Note that there are no numbers on the graph. To find your 5-number summary, press $ and use the left and right keys to switch between values. Write these numbers in the space below.

______

Min. 1st Q. Med. 3rd Q. Max.

  1. An interesting way to examine data is to place two different plots on the same screen. Press`! to return to the STAT PLOTS screen. This time choose “2:Plot2." Turn On the plot and select the histogram icon („). Finally, make your Xlist be Ω. Press graph to view your new plot.
  1. Use $ to discover some of the values from the two plots. The left () and right keys () will guide you to the different values within a plot, and the up (:) and down (;) keys will allow you to switch between the two plots.
  1. Which plot do you think best describes the data? Why?

9. Repeat steps 3-8 with æ. Be sure to replace Ω in the plots. We don’t want to view Ω and æ simultaneously, at least not yet. Answer the following questions for æ.

(a)Find the 5-number summary for æ.

______

Minimum 1st Q. Median 3rd Q. Maximum

(b)In comparing the histogram for æ with its boxplot, which plot do you think best describes the data? Why?

  1. Now you are going to compare the boxplots of Ω and æ. First redo step 3 by having Plot1 show the boxplot for Ω. Next change Plot2 to make it display a boxplot of æ. When you are done, your STAT PLOTS screen should look something like the screen on the right. Press % to view the results.
  1. The simultaneous boxplots allow you to compare the shape or distribution of each data set. Describe the shape of each and compare the shape of Ω with æ.
  1. Based on the above analysis, do you think the two class scores are equivalent or not? Use numbers and descriptions to back up you assertions.
  1. While in the STAT PLOTS sub-menu, did you notice this: fl?? It looks like a boxplot, but it must have different properties to warrant its own icon. To discover what these are you will have to change the data in æ. We don’t want to corrupt the data, so we will transfer a working version to ø. Press Se to go to your lists. Go over to ø and press : to highlight ø. In the space type æ. Now you have a working copy of æ.

14. Go down to the bottom of ø at L3(21). Enter into the column two more values: 90 and 100.

Press `! to go to the STAT PLOTS screen. Choose “3:Plot3" by pressing e. Turn On the plot by pressing e. Now, in the Type row, choose fl. Press % to view the results.

  1. a. What happened when the new points 90 and 100 were added?

b. Describe how this new plot is different from the original.

c. An outlieris a point that “does not ‘fit in well’ with the rest of the data.” Regarding the new æ, which point(s) (if any) are outliers?