Prepared by Krystyna Karminska-Makagon, TNCC, Virginia
Problems:
Given a set of data.
1. Find the mean, median, standard deviation, the first and the third
quartile, and the range.
2. Find the box graph and
the histogram.
Example: the following exercise
comes from the textbook Elementary Statistics by Mario Triola, the seventh edition. (Review Exercises, page 113)
“The amounts of time spent on paperwork in one day were obtained from a
sample of office managers with the results given below (based on data from Adia Personal Services):
3.7 2.9 3.4 0.0 1.5 1.8 2.3 2.4 1.0 2.0
4.4 2.0 4.5 0.0 1.7 4.4 3.3 2.4 2.1 2.1”
Solution:
First,
we will make a list of data.
1.
Press STAT, then 5 (set up the editor); ENTER.
2.
Press STAT, ENTER to start editing your list.
Type in your data.
Press ENTER after every entry. Data will be listed under L1.
Press 2nd QUIT when done.
Next
we will sort the list in
ascending order to observe the mode. Press STAT, 2:SortA
to display the command on the home screen. Press 2nd, 1 to write L1. Press
ENTER. The calculator will answer: “done”. Go to STAT, EDIT to see the list.
Scroll the list and observe the repeating values. The set is multimodal:
several numbers are listed twice.
Now
we can obtain the statistics of
the sample:
Press
STAT, highlight CALC, then choose 1: one-variable
statistics. Follow with ENTER.
You
will see on the screen the following:
`x =2.395 (the mean)
3x=47.9 (sum of all data)
3x2=146.53 (sum of all squares)
Sx=1.293902705 (standard deviation of the sample)
sx=1.261140357 (standard deviation of the
population)
n=20 (sample
size)
¯
The
down arrow indicates that there is more to read on. Scroll the screen by
pressing Ñ (Down Arrow) a few times. You’ll see:
minX=0 (the minimum)
Q1=1.75 (the first quartile)
Med=2.2 (the median: the second
quartile)
Q3=3.35 (the third quartile)
maxX=4.5 (the maximum)
The
range can be found by subtracting the minimum from the maximum value.
The
last five numbers above represent five number summary
needed to draw a box graph.
First,
make sure that all graphs in the area “Y=” are either erased or unselected.
We
are going to draw two graphs.
A.
The box graph: (“Box-and-whiskers” graph)
Let’s
find the box plot for our sample. Press 2nd, then “Y=” to open STAT PLOTS menu.
Press 1 to choose plot 1. Press ENTER to turn it on. Press Down
Arrow, then Right Arrow (three or four times) to choose a type of the graph.
You may choose either the first box, or the second one. The difference will
occur if your sample contains any “outliers” (the values that significantly
differ from the other data). The first box will display the outliers as
separate points. The second box will display the whiskers, which is a line
connecting minimum and maximum value. Highlight the box of your choice, press
ENTER. Ready to graph? Press ZOOM and 9 for ZoomStat. Press TRACE and move the cursor to the left or to
the right to read the minimum, Q1, Q2, Q3 and the maximum value.
B. The histogram:
Go
to STAT PLOT, choose again Plot 1; use the Down and Right Arrow and ENTER
button to choose the symbol of histogram (press the right arrow twice). Press ZoomStat (ZOOM 9). If you press TRACE, the lower and upper
limit of the first class will be displayed on the left side of the screen and
the frequency n=number will show up on the right side. Move the cursor to the
right and observe the changes in frequency. If you want to change the width of
the classes, go to WINDOW and change Xscl, that is,
the class width. If you choose smaller width, you will produce a larger number
of classes (bars). Try to change the width now. The calculator chose xscl=0.9 and drew 6 classes. Try xscl=1,
press GRAPH. Then try xscl=2, then xscl=0.5. You can’t go very far with narrowing your class
width: for instance, xscl=0.1 will produce an error.
Problems:
1. Find the scatter plot
2. Find the regression line
3. Find the correlation
coefficient.
Example:
This
time we will solve a problem from the textbook Algebra and Trigonometry
by Larson, Hostetler and Edwards, the second edition, page 229.
“On Friday 22 students in class were asked to keep a track of the hours
they spent studying for a test on Monday. The data are shown below. Construct a
scatter plot [...] then determine whether the points are positively correlated,
negatively correlated or have no discernible correlation.”
The
following data are copied from the text: x represents the number of hours and y
represents the score on the test.
|
x |
0 |
1 |
2 |
3 |
3 |
4 |
4 |
5 |
5 |
5 |
5 |
6 |
6 |
6 |
7 |
7 |
8 |
8 |
|
y |
40 |
41 |
51 |
58 |
49 |
48 |
64 |
55 |
69 |
58 |
75 |
68 |
63 |
93 |
84 |
67 |
90 |
76 |
Solution:
We
will start with clearing the lists
and typing in the new sets of data.
Press STAT and choose EDIT by pressing ENTER. The lists of data are displayed
on the screen. Press Up Arrow to highlight the name
L1. Then press CLEAR followed by ENTER. The set you wrote before has
disappeared. Type in the first row of data (the x values)
under L1 and the second row (the y values) under L2. When you finish,
compare the numbers of entries in each set: they should match.
Now
let’s make a scatter plot for
the set of paired data (x,y).
Press 2nd, STAT PLOT, then ENTER to choose Plot1, press ENTER to set
it on, Down Arrow to highlight the scatter graph and ENTER to choose it. By
default, your X-list will be set on L1 and y-list on L2. If you wanted to store
your data under different list names, you will now have to highlight the
positions that require changes, then pick the
appropriate names from the LIST menu. If you kept your entries in the lists L1
and L2, then you don’t have to change anything now.
Press
Zoom Stat (ZOOM 9) to plot the points (x,y)
on the screen.
To
obtain two variable statistics press STAT, then the Right Arrow, for
calculations submenu (CALC), then choose 2: 2-var stats, ENTER. (Note: if you renamed
your xlist or ylist, you
have to use those names following 2-var stat command, and separated by a comma;
for instance, 2-Var Stats HOUR, SCORE).
Have
you already noticed that the points on the scatter plot are forming a straight
line? The line that best fits a collection of points is called the regression
line. Let’s find the equation for that line and the correlation coefficient.
The correlation coefficient is a fraction: a number between –1 and 1. It
reflects how closely the sets x and y are related. The greater the absolute
value of the coefficient is, the more significant the correlation between two
sets of data.
In
order to find regression line press
STAT, go to CALC and choose 4:LinReg (ax+b), ENTER.
Note:
if you have more than two lists of data, you will have to type the names of the
two lists you work with now; for example: RegLine(L3,L4), or RegLine(HOURS, SCORE).
You
may draw the line by pressing
Y=, then VARS, 5:Statistics. Press Right Arrow twice
to highlight EQ (equations), and press ENTER to execute 1: RegEQ.
The formula for regression line will be automatically pasted to Y1=.
Press
GRAPH to display the line and the scatter points. Press TRACE, then Up or Down Arrow to move between the graph and the scatters.
Move along the graph with Right or Left Arrow. Now you can answer the questions
about some “predictions”, such as how long a student would have to study to
score 100. Draw the line y2=100 and press 2nd CALC, choose 5, ENTER,
ENTER, ENTER to find its intersection with the
regression line. Or you can simply trace the regression line, and zoom in
several times around the point whose y-coordinate is 100. In both cases make
sure to change x-max in the window range to bring the solution point onto the
screen.
(By
the way - the answer is 11 hours.)
Correlation coefficient is not displayed
automatically. We have to set the diagnostic display mode. First, quit the
graphing screen by pressing 2nd and QUIT. Now, go to the catalog:
press 2nd, 0, press D (the key with x-1) and scroll down
the screen to mark Diagnostic On. Press ENTER, ENTER. The calculator responds
with “Done”. Bring the formula for the regression line again
onto the screen by pressing STAT, CALC, 4, ENTER. The display is:
LinReg
y=ax+b
a=5.708941027
b=36.87444515
r2=.6896321709
r=.8304409497
The
last number, r, is the correlation coefficient and r2 is the
coefficient of determination.
The
TI-83 can do much more than diagnosing and plotting the sets of data.
It
lets you play with the lists: you can fill them with random numbers, build them
with algebraic formulas, or perform math operations on the entire lists.
It
provides you with tools to perform a statistical test or find the intervals of
confidence; it will also draw the results of the tests. It plots normal
distribution and computes the normal distribution probability. It knows
Student-t distribution and much more. For more information refer to your
owner’s manual.
Reference:
TI-83 Graphing Calculator Guidebook, Texas Instrument Inc., 1997