# MATH200A Program —

Basic Statistics Utilities for TI-83/84

Copyright © 2008–2017 by Stan Brown

Basic Statistics Utilities for TI-83/84

Copyright © 2008–2017 by Stan Brown

Stan Brown’s MATH200 classes will use this program, but any statistics student should find it helpful.

**See also:**
Troubles? See TI-83/84 Troubleshooting.

**See also:**
Advanced students might also want to download the program
described in MATH200B Program — Extra Statistics Utilities for TI-83/84.

`MATH200A`

Program OverviewBecause this program helps you,

please click to donate!Because this program helps you,

please donate at

BrownMath.com/donate.

please click to donate!Because this program helps you,

please donate at

BrownMath.com/donate.

The program is in two parts, MATH200A and
MATH200Z. You need both on your calculator, even though
**you won’t run MATH200Z directly.** It works with
all TI-83 Plus calculators and all TI-84 calculators, including the
color models.

**If you have a “classic” TI-83**, not a Plus or Silver,
follow the directions below but put M20083A and
M20083Z on your calculator, not MATH200A and
MATH200Z. (M20083A and M20083Z aren’t
being updated after version 5,
which was released in August 2012, so you will see some differences
from the screen shots in this document.)

There are three methods to get the programs into your calculator:

- If you have a TI-84, download MATH200A.zip (111 KB, updated 28 Dec 2016), and unzip it. Use the USB cable that came with your calculator, and the free TI Connect CE software from Texas Instruments, to transfer the MATH200A.8XP and MATH200Z.8XP programs to your calculator.
- If a classmate has the programs on her calculator
(any model TI-83/84),
she can transfer them to yours, provided you both have a USB port
or you both have a round I/O port. Connect the appropriate cable to
both calculators, inserting each end firmly.
On your calculator, press [
`2nd`

`x,T,θ,n`

*makes*`LINK`

] [`►`

] [`ENTER`

]. Then on hers press [`2nd`

`x,T,θ,n`

*makes*`LINK`

] [`3`

], select MATH200A (or M20083A; see above), and finally press [`►`

] [`ENTER`

]. If you get a prompt about a duplicate program, choose Overwrite.Repeat for MATH200Z (or M20083Z; see above).

- Or, as a last resort, key in the programs. See MATH200A.pdf, MATH200Z.pdf, and MATH200A_hints.htm in the MATH200A.zip file.

Press the [`PRGM`

] key. If you can see `MATH200A`

in the menu, press its number; otherwise, scroll to it and press
[`ENTER`

]. When the program name appears on your home screen,
press [`ENTER`

] a second time to run it. Check the splash screen to make sure you have the
latest version (v7.2), then press
[`ENTER`

].

The menu at right shows what the program can do:

`Histograms etc`

: make a histogram or polygon, or overlay both, for a frequency distribution, a relative frequency distribution, a simple list of numbers, a binomial probability distribution, or a generic discrete probability distribution.`Box-whisker`

: plot a box-whisker diagram, showing any outliers`Binomial prob`

: compute probability or cumulative probability of a binomial distribution`Normality chk`

: test whether your sample is drawn from a normal distribution`Sample size`

: find the necessary sample size for a given confidence level and margin of error for binomial data (one or two populations) or numeric data (with population standard deviation σ known or unknown)`GoF test`

: test goodness of fit for categorical data in one population`Two-way table`

: test independence or homogeneity for categorical data in a two-way table

If you ever need to break out of the program
before finishing the prompts, press [`ON`

] [`1`

].

If you run the program on a TI-84 with a higher-resolution screen, some displays will look slightly different, but all keystrokes will be the same.

The program is protected so that you can’t edit it accidentally. If you want to look at the program source code, see MATH200A.PDF and MATH200Z.PDF in the downloadable MATH200A.ZIP file.

Each procedure leaves its results in variables in case you want to use them for further computations. For details, please see the separate document MATH200A Program — Technical Notes.

Making a histogram or
frequency polygon using native TI-83/84 commands is kind of tedious,
especially setting up the WINDOW screen.
This part of the `MATH200A`

program automates the process.
The program can create a
**histogram or polygon, or both in overlay**, for these
distributions:

- grouped distribution (discrete or continuous) with frequencies, relative frequencies, or probabilities
- simple list of numbers (discrete or continuous)
- ungrouped distribution (discrete data) with frequencies or relative frequencies
- general discrete probability distribution
- binomial probability distribution

To use the program, put the class midpoints or ungrouped data
in a statistics list and the frequencies (if applicable) in another.
(Some people use the term
**class marks** instead of **class midpoints**; they mean the
same thing.)

Then press [`PRGM`

], select
`MATH200A`

, and press
`1:Histograms etc`

. The program will prompt you for the
necessary information and will check
silently to make
sure your inputs are in valid form. Then it will ask whether you want a
histogram, polygon, or both, and will produce your desired graph. (The
program uses an algorithm to ensure that there are an appropriate
number of dots vertically on the screen.)

**Restrictions**:
If you have frequency data, the data list and
frequency list must be the same length and the class widths in the
data list must all be the same; the program checks this. The TI-83 and
TI-84 won’t let you make a histogram with more than 47 classes,
and the program also checks this. Finally, the program also
won’t let you make a grouped frequency histogram with just one
or two classes, because that’s silly.

Class Boundaries | Class Midpoints | Frequency |
---|---|---|

20 ≤ x < 30 | 25 | 34 |

30 ≤ x < 40 | 35 | 58 |

40 ≤ x < 50 | 45 | 76 |

50 ≤ x < 60 | 55 | 187 |

60 ≤ x < 70 | 65 | 254 |

70 ≤ x < 80 | 75 | 241 |

80 ≤ x < 90 | 85 | 147 |

The grouped frequency distribution at right shows the ages reported by Roman Catholic nuns, from Johnson & Kuby 2004 [full citation at http://BrownMath.com/swt/sources.htm#so_Johnson2004], page 67. Show the data as a histogram and as a frequency polygon.

**Solution**:
Your class marks (class midpoints) are 25, 35, up through 85, and your class
width is 10. On the `STAT EDIT`

screen, enter those
class midpoints in one list and the frequencies in another.
Here the class midpoints have been entered in L5 and the
frequencies in L6.

Run the `MATH200A`

program and select `1:Histograms etc`

.
On the Data Arrangement screen, select [`3`

] for grouped
frequency distribution.
Specify your list of class midpoints (class marks) and your frequency
list.

Next, select which plots you want. In the illustration,
I’m selecting both plots. If you’re not doing frequency
polygons in your class, select [`1`

] for just the histogram.

The output is shown at right, once with the histogram and polygon on the same screen, and once with just the histogram.

You can **trace the histogram** by pressing
[`TRACE`

].
This lets you see the class boundaries and number of data points in each
class.

Press [`◄`

] and [`►`

] to
move through the classes. To suppress the tracing
information, press [`GRAPH`

] again.

To **trace the polygon**, if you selected both, press
[`TRACE`

] [`▼`

].
This lets you see the class midpoints and number of data points in each
class, instead of the class boundaries.

Why press [`▼`

]? When you press
[`TRACE`

], the calculator starts with a trace of Stat Plot 1.
The up or down cursor key moves between plots. Since the frequency
polygon is Stat Plot 2, you are tracing it when you see P2 in the
upper left corner, as shown in the illustration.

11 | 15 | 14 | 12 |

9 | 8 | 7 | 5 |

6 | 11 | 10 | 10.5 |

12 | 11 | 13 | 2 |

6 | 4 | 13.5 |

Suppose you want to make a histogram of the class performance on a 15-point quiz, where the scores are shown at left. You don’t want to bother to group these 19 scores by hand, so you enter them in a statistics list such as L1 and let the program do the grouping for you. (An alternative graphical display is the box-whisker plot.)

Run the `MATH200A`

program and select `1:Histograms etc`

.
When the program asks your data arrangement, select
[`1`

] for a plain list of numbers.

Then enter the name of the list that contains your numbers. For a plain list of numbers, the program needs to know how you want to group your data, so it asks you to specify the lower bound of the first class as well as the class width. Since 0 is the lowest possible grade, that’s the obvious lower bound for this example. The quiz has a possible maximum score of 15, and 10% of that is 1.5 points. This is a good class width because grades of D, C, B, and A will each be one bar of the histogram.

What if you’re not sure which class width to use? Try several class widths, one at a time, and see what the histograms look like. You can pick the class width that seems to display the shape of the data most clearly.

At this point you may see a pause as the program computes the number of classes and places each data point into a class. (It uses statistics lists LD for the class midpoints (class midpoints) and LF for the computed frequencies.)

For a simple list of numbers, the program will make only a histogram, not a frequency polygon.

In first looking at that histogram,
you might think there are four Ds, three Cs, two Bs,
and one A. But when you check this by pressing [`TRACE`

] you see
that’s not correct. The highest class is the 15 to 16.5 class,
since someone had a perfect score of 15. (Remember: when a value
is right on a class boundary, it is always assigned to the higher
class.) So the top *two* classes in the histogram represent As
(three students), the next lower is the three Bs, the next lower
(shown at right) is four Cs, the next lower is two Ds, and the rest
are Fs.

Children per family | Freq. |
---|---|

0 | 9 |

1 | 6 |

2 | 10 |

3 | 2 |

4 | 2 |

5 | 1 |

You have recorded the numbers of children in 30 randomly selected families that used a community center in a given week, and you want to show a picture of the discrete distribution. There are only a few different values (numbers of children per family), so you choose an ungrouped frequency distribution.

Enter the data in two lists such as L3 and L4, run
the `MATH200A`

program, and select `1:Histograms etc`

.
For Data Arrangement, select [`2`

], ungrouped distribution.
Enter your data list and frequency list as usual.

When prompted, select whether you want a histogram, a frequency polygon, or both. I’ve selected a histogram, and the results are shown below.

The vertical line in the histogram is the y
axis — you can remove it, if you want, by pressing
[`2nd`

`ZOOM`

*makes* `FORMAT`

] and selecting `AXES`

`OFF`

. Also in the
histogram, notice that the vertical rows of dots run through the
center of each bar rather than along the edges. This reminds you that
you should label the bars of an ungrouped frequency distribution under
the centers, not the edges as you would label a grouped histogram.

If you want to trace the ungrouped frequency histogram, follow the same procedure as for tracing a grouped frequency histogram.

If you have a relative frequency distribution or probability distribution, you plot it in almost the same way as a frequency distribution. The main difference is that the relative frequencies or probabilities must add up to 1, and the program checks this for you.

(There’s a special case: the binomial probability distribution. For this, please see Histogram of a Binomial Distribution below.)

Number of dice alike | Probability |
---|---|

1 | 720/7776 |

2 | 5400/7776 |

3 | 1500/7776 |

4 | 150/7776 |

5 | 6/7776 |

Here’s an example of a general discrete probability distribution, drawn from the rainy-day game Yahtzee. In Yahtzee you roll five dice and try to make various combinations. Shown at right are the probabilities for number of dice alike when you roll five standard six-sided dice. (Thanks to Paul Sperry for help with the probabilities.) You can make a histogram of this probability distribution.

Notice, by the way, that you’re more than twice as likely to roll three of a kind as to roll “none of a kind” or all five different: P(3) = 1500/7776, and P(1) = 720/7776. And you’re over seven times as likely to roll two of a kind (either two the same and three all different, or two pairs with the fifth die different): P(2) = 5400/7776. Of course, Yahtzee isn’t just about the initial roll. You get two tries to improve your combinations by re-rolling some of your dice.

As before, put the x’s (1–5 this time) in one list and
the p’s in another. Run the `MATH200A`

program and select `1:Histograms etc`

.
The data arrangement this time is a
probability distribution, so specify [`4`

] and then enter your
data list and probability list.

The finished probability histogram is shown at right. (For a probability distribution, the program automatically makes a histogram, with no option to make a polygon.) Notice that the bar for two of a kind is much higher than any of the others.

If it wasn’t obvious from the numbers, you can see
from the histogram that this distribution is skewed right.
If you like, you can press the [`TRACE`

] button and see
the numerical value of the probability for each outcome. For example,
P(2) = 0.6944, meaning that when you roll five dice you have
almost a 70% chance of getting two of a kind.

What about two of a kind *or better*?
This is a classic “at least” problem, and the complement is
your friend.
P(1) is 0.0926 and 1−P(1) = 0.9074. You have better
than a 90% chance of getting at least two of a kind when rolling five
dice.

A later section will show how to
compute the binomial probability of a particular x or a range of x.
But for an **overview of a distribution**, a histogram is most
helpful. This part of the program creates one for you.

**Example**: Suppose 65% of registered voters in Dryden
are Republicans. Plot the binomial probability histogram for the
random variable “number of Republicans in a random sample of ten
registered voters“.

**Solution**: Unlike the other cases, for the binomial probability
histogram you don’t enter anything in statistics lists. Instead,
the program will prompt you for the necessary numbers.
Run the `MATH200A`

program and select `1:Histograms etc`

.
When the program asks your data arrangement, select `5`

for
a binomial probability distribution.

The program will then ask for number of trials (n) and probability of success (p). You can select a histogram or polygon as usual; the histogram is shown at right.

**Interpreting the histogram**:
The bars are each one unit wide, running from 0 to n (0 to 10, for
this example), with the dots marking each integer and the y axis
marking the minimum possible value of successes, 0.
You can see that the most likely result for a sample of
ten registered voters is seven Republicans, with six being just
slightly less likely. Eight and five are next most likely, then nine
and four. A sample of ten is quite unlikely to
contain ten or three Republicans, and fewer than three Republicans are
extremely unlikely.

To see the probabilities as numbers, trace the histogram, or use Probability of a Binomial Experiment to compute them.

**Summary:**
A modified box-whisker plot is a
quick graphical representation of a data set. It plots the
**five-number summary** (minimum, first quartile, median, third
quartile, maximum) and also shows **outliers** if the data set
contains any. The `2:Box-whisker`

part of the `MATH200A`

program makes a modified
box-whisker diagram for one data set or compares two or three data
sets by stacking box-whisker plots.

11 | 15 | 14 | 12 |

9 | 8 | 7 | 5 |

6 | 11 | 10 | 10.5 |

12 | 11 | 13 | 2 |

6 | 4 | 13.5 |

Here again is the set of quiz scores. You’ve already seen them graphed as a histogram, but a box-whisker plot is another way to get a sense of the shape of the data.

Enter the data in any statistics list and run the `MATH200A`

program.
Select `2:Box-whisker`

and you see a prompt for the number of samples. You
can make a box-whisker plot of a single data set, or you can
compare two or three data sets. The
quiz scores are a single sample, so you choose [`1`

].

Then the program asks whether you have a plain list of numbers
or an ungrouped frequency distribution.
Here you have a plain list of numbers, so you choose data arrangement
[`1`

].
Finally, the program asks you for your data list.

**Caution**:
**Never make a boxplot of a grouped frequency distribution.**
Only a simple list of numbers or an ungrouped
frequency distribution is suitable for a box-whisker plot.
If you have a grouped frequency distribution,
a box-whisker plot won’t be accurate and you should be using a
histogram.

The box-whisker plot now appears (below left). You can see at a glance that it has no outliers, and that it’s slightly skewed left.

You can also trace the box-whisker, to see the five-number
summary and the values of any outliers. Press the [`TRACE`

] key
and then use [`◄`

] and [`►`

] to move
left and right. In the illustration below right you can see that the
median quiz score was 10.5.

11 | 15 | 14 | 12 |

9 | 8 | 7 | 5 |

6 | 11 | 10 | 10.5 |

12 | 11 | 13 | 2 |

6 | 4 | 13.5 | 22 |

What would an outlier look like on a box-whisker plot? Any outliers show up as isolated points separate from the main diagram.

For example, take the same
set of data, but append a twentieth data point, 22. Now make the plot
and you’ll see the result at right. The [`TRACE`

] key and arrow
keys will display the values of any outliers as well as the
five-number summary.

Sullivan 2011 [full citation at http://BrownMath.com/swt/sources.htm#so_Sullivan2011], page 168, shows data for two groups of rats. One group was sent into space; the control group was treated the same except for the space flight. Their red blood cell mass was measured in milliliters.

Flight | Control |
---|---|

8.59 6.87 7.00 6.39 7.43 9.79 9.30 8.64 7.89 8.80 7.54 7.21 6.85 8.03 | 8.65 7.62 7.33 7.14 8.40 8.55 9.88 6.99 7.44 8.58 9.14 9.66 8.70 9.94 |

Plotting the two data sets as two boxplots on the same screen is a good way to get a sense of whether there is a difference between the samples.

(By the way, these two samples happen to be the same size, but you can just as well stack boxplots with samples of different sizes.)

Put the flight group in one statistics list and the control group in
another. Then run the `MATH200A`

program, select `2:Box-whisker`

, and select
`2:Compare 2 smpl`

.

For each sample you’ll be asked your data arrangement. (That lets one sample be a plain list and the other an ungrouped frequency distribution, but in this example both are plain lists.) Enter the names of the two lists, as shown below right.

The results are shown at right. You can see that the flight group, as a group, had lower blood-cell mass than the control group, even though some individuals from the two groups had equal blood-cell mass. Look particularly at the medians: the median for the control group is about equal to the third quartile of the flight group, meaning that about three quarters of the flight rats had blood-cell mass lower than the median of the control group.

Is that enough to say that space flight lowers blood-cell mass
in rats in general? Not yet. Later in the course, you’ll learn
how to use a two-sample t test to tell whether space flight
lowers blood-cell mass in rats — whether there is a
difference between the *populations* of all space rats and all
earthbound rats.

**Summary:**
If you have a fixed number of trials
*n*, and each trial has only two outcomes (called success and
failure), and the probability of success *p* is the same on each
trial, then you have a binomial probability distribution. This
part of the program computes the probability of a specific number of
successes or the probability of a range of numbers of successes.

**TI-89 users:**
Please see
Binomial Probability Distribution on TI-89.

**See also:**
You can also make a Histogram of a Binomial Distribution.

The program always asks you the number of trials *n*,
the probability *p* of success on one trial, and the number of
successes *from* and *to*. If you want a specific number of
successes rather than a range, then *from* and *to* will be
the same number.

**Example 1:** Larry’s batting average is .260. If he’s
at bat four times, what is the probability that he gets exactly two
hits?

**Solution**:

n = 4, p = 0.260, x = 2

Note: Some textbooks use r for number of successes, rather than x.

Here you want the probability of exactly two successes, so
`FROM`

and `TO`

will both be 2. In other words, you
are computing the probability of 2 through 2 successes.
Run the `MATH200A`

program, select `3:Binomial prob`

, and specify

n = 4, p = .26, from = 2, to = 2.

The input and output screens are shown at right.

Answer: P(2) = 0.2221

**Caution:**
Sometimes the probability is very
small and the calculator reports it in scientific notation, such as
5.4189E-6. The exponent is not a decoration, and you must not report
the probability as simply 5.1489. Probabilities are never greater than
1!

Conventionally, we
**round probabilities to four decimal places.** If the probability
is smaller than 0.0001 (smaller than 1E-4), you can show it to two
significant figures, such as 1.3×10^{-6}, or report it as
<0.0001. Or you can convert it to a decimal,
5.4189E-6 → 0.000 005.

**Example 2:** Larry’s batting average is .260. If he’s
at bat four times, what is the probability that he strikes out all
four times?

**Solution**: Four out of four strikeouts means no hits,
so you’ll put these values into the program:

n = 4, p = 0.260, from = 0, to = 0

The output screen is shown at right.

Answer: P(0) = 0.2999

**Example 3:** Suppose 65% of the registered voters in Dryden are
Republicans. In a random sample of ten registered voters, what’s
the probability of fewer than six Republicans?

**Solution:** “Fewer than six” is
*zero through five*.

n = 10, p = 0.65, 0 ≤ x ≤ 5

Run the `MATH200A`

program, select `3:Binomial prob`

, and enter

n = 10, p = .65, from = 0, to = 5

Answer: 0.2485

There’s about one chance in four of getting fewer than six Republicans in a random sample of ten registered voters.

By plotting data on your TI calculator, you can easily see
how close they are to a normal distribution. The special
**quantile plot** or **normal probability plot** asks what the
distribution would look like if it were normal, and plots that against
the actual distribution. The closer the points seem to be to
a straight line, the more nearly normal the original distribution.

The most common
**application is in inferential statistics**: with a small sample
(less than about 30), you need to make sure that the population is
normally distributed before you perform a Student’s
t test.

**Example**: Consider these vehicle weights (in pounds):

2500, 3250, 4000, 3500, 2900, 4500, 3800, 3000, 5000, 2200

Construct a plot to decide whether these vehicle weights seem to be normally distributed.

**Solution**: Put the data in any statistics list,
then press [`PRGM`

], scroll down to `MATH200A`

, and
press [`ENTER`

] twice. Select `4:Normality chk`

.

The program will make the plot and display the correlation coefficient r and sample size n, as shown at right. If the points are clearly linear (close to a straight line), r is close to 1 and you know that the sample was nearly a normal distribution; if they are clearly not near a straight line, r is far from 1 and you know that the sample is non-normal.

How far from r=1 is too far? The program also
displays “critical r”, from
How Normal Probability Plots Work.
**If r≥crit, it’s close enough to 1**;
**if r<crit, it’s too far from 1**
and the data are not
normally distributed. (That may not be the end of your analysis. Some
statistical procedures, such as a t test, are still valid for
*moderate* departures from normality.)

For this example, the program computed a correlation coefficient of 0.9936 and critical value of 0.9179 for n=10. If it weren’t already obvious from the plot, r≥crit would tell us that the original data are approximately normal.

If you’re using M20083A rather than MATH200A,
you have to compute the critical number yourself, with this formula:
1.0063−.6118/*n*+1.3505/*n*²−.1288/√*n*.

**Summary:**
Before you start gathering data, you plan for how large a
sample you will need. This depends on your desired margin of error
and confidence level, and on the type of the data and your prior
estimates. This part of the `MATH200A`

program pulls all this together for you
for the most common cases.

**See also:**
How Big a Sample Do I Need? gives the statistical concepts
with examples of calculation “by hand”.

When you run the `MATH200A`

program and select
`5:Sample size`

, you are prompted first for the type of data, then for the
margin of error, prior estimate, and confidence level.
The program can find the
necessary sample size for binomial data for one or two populations, or
for numeric data for one population whether the population standard
deviation σ is known or not. (At present the program
can’t find sample size for numeric data for two populations.)

**Caution:**
There are other criteria for sample
size. For example, with numeric data your sample should be at least 30
unless you know that the population is normally distributed. Always
**check that your sample will meet the requirements before you begin gathering data.**

The key here is whether you know the standard deviation of the population or not. If you don’t know σ, as usually you don’t, then you use s but the computation of sample size is different

**Example 1—numeric data with known σ**.

You want to estimate the average hourly output of a
machine to within ±1.5, with 90% confidence. Based on
historical data, you have reason to believe that the standard
deviation of the machine’s hourly output is 6.2. How large a
sample do you need?

**Solution:** Note first that this is
**not a realistic situation. It’s pretty unlikely** that you
would know the standard deviation of a population but not know the
mean of that population. However, statistics texts always begin with
this case because it’s the simplest way to demonstrate the
principles. You leave Perfectland and enter Realityville in the other
cases. With that said—

Marshal your data: 1−α = .90,
E = 1.5, and σ = 6.2.
Run the `MATH200A`

program and select
`5:Sample size`

, then `1:Num known σ`

. Enter σ=6.2,
E=1.5, and C-Level=.9 or 90.
The output screen echoes back your inputs and tells you the
critical z for that confidence level as well as the minimum necessary
sample size.

**Conclusion**: If the standard deviation of the
population is 6.2, then to get a 90% confidence interval about the
population mean μ with a margin
of error no greater than 1.5, you need a sample of at least 47.

**Example 2—numeric data with unknown σ**

This is **the realistic case** for estimating a population
mean. Usually you don’t know the standard deviation of the
population, and you make a small pilot study to estimate it or you use
a prior estimate.

The `MATH200A`

program uses the method shown in
Sample Sizes Required
in NIST/SEMATECH e-Handbook of Statistical
Methods.

Here’s a modified form of the previous example. You want to estimate the average hourly output of a machine to within ±1.5, with 90% confidence. A small pilot study finds a sample standard deviation of the machine’s hourly output is 6.2. How large a sample do you need?

**Solution**:
Marshal your data: 1−α is .90,
E is 1.5, and s (not σ) is 6.2.
Run the `MATH200A`

program and select
`5:Sample size`

and then `2:Num unknown σ`

.
Enter s and E, then .9 or 90 for C-Level as before.
There may be a delay, because the program must compute one
or more inverse t numbers.
The output screen echoes back your inputs and tells you the
critical t as well as the minimum necessary
sample size.

**Conclusion**: If you don’t know the standard
deviation of the population, but you have a prior sample with a
standard deviation of 6.2, then to get a 90% confidence interval about
the population mean μ with a margin of error no greater than 1.5,
you need a sample of at least 49.

Why the difference from the previous example? In the previous case you knew the population standard deviation, and here you don’t. Since an additional parameter is unknown, it takes a larger sample to get the same level of precision for an estimate of μ. Another way to look at it: with unknown σ, you’re using the t distribution, which is more spread out than the z (normal) distribution, and therefore t needs a lager sample to achieve the same margin of error.

The program helps you find the sample size for estimating a population proportion p (some books use π) for binomial data. Use a prior estimate p̂ in this computation if you have one; if you have no idea of the population proportion then use 0.5.

**Example 3—binomial data for one population, with prior estimate**

What percent of the voters would vote for your
candidate if the election were held today? You want 95% confidence in
your answer, with a margin of error no more than 3.5%. Last
month’s poll showed your candidate had 42% support. How many
voters do you need to survey?

**Solution**: Here you have 1−α = 0.95,
E = 0.035 (careful! not 0.35), and p̂ = 0.42.
(When you have a prior estimate, use it; otherwise use 0.5.)
Run the `MATH200A`

program and select
`5:Sample size`

, then `3:Binomial`

.
Enter your prior estimate, your margin of error, and your confidence
level, and you get the output screen shown at right.

**Conclusion**: If the true population proportion is
somewhere in the neighborhood of 42%, and you want a 95% confidence
interval with a margin of error no more than 3.5%, you’ll need
a sample of at least 764.

**Example 4—binomial data for one population, with no prior estimate**

Suppose you didn’t have any idea of the proportion of the
population that planned to vote for your candidate? In that case,
you’d use 0.5 for your prior estimate, and you’d have the
output shown at right. The necessary sample size would rise to
784.

With binomial data for two populations, you compute necessary sample size to estimate the difference between the two proportions. This is different from the sample size needed to estimate the proportion in either population on its own.

It’s not actually necessary to have the two sample sizes equal, but that’s the only way it’s possible to compute them when neither one is specified up front. Just as before, use prior estimates for the population proportions if you have them, and otherwise use 0.5.

**Example 5—binomial data for difference of two population proportions, with prior estimates**

Suppose you’d like to know how your candidate’s
support differs between men and women. You know that overall support
is 42%, and you think the candidate is 10 percentage points more
popular among women versus men. How many of each sex must you
survey to answer the question with 95% confidence and a margin of
error no more than 3½%?

Do you have an estimate of p_{1} and p_{2}?
Yes, since the overall support is 42% you expect that men’s and
women’s support is not too different from that. If women are 10%
higher, then you estimate women at 47% and men at 37%.)

**Solution**: Here you have 1−α = 0.95,
E = 0.035, p̂_{1} = 0.47, and
p̂_{2} = 0.37.
Run the `MATH200A`

program and select
`5:Sample size`

, then `4:2 pop binomial`

.
Enter your prior estimates, your margin of error, and your confidence
level, and you get the output screen shown at right.

**Conclusion**: If the true population proportions are
somewhere in the neighborhood of 47% and 37%, and you want a 95% confidence
interval on the difference with a margin of error no more than
3½%, you’ll need a sample of at least 1513 women and 1513
men.

**Example 6—binomial data for difference of two population proportions, with no prior estimates**

And suppose this was the first poll and you had no idea of your
candidate’s support among women and men? In that case you use
0.5 for both p̂_{1} and p̂_{2}, with the results shown at right.
With no prior estimate, to get a 95% confidence interval on the
difference in support, with only a 3½% margin of
error, you’d need 1568 women and 1568 men.

**Summary:**
The goodness-of-fit or GoF test determines
**how well a multinomial model (more than two categories) matches the sample data**.
This part of the `MATH200A`

program computes the χ² test statistic and the
p-value and graphs the distribution. (Though the TI-84 has a GoF test,
it still makes you do part of the computation by hand.)

**TI-89 users:**
Please see
Testing Goodness of Fit on TI-89.

To perform a goodness-of-fit test, first
**put your model in L1 and your observed numbers in L2.**
Caution: the model is usually percentages or ratios, but the
observed numbers are always the actual counts in whole numbers, never
percentages.
Then run the `MATH200A`

program and select `6:GoF test`

.
The program will ask you to confirm
that you’ve filled the two lists, and then it will perform all
computations for the χ² goodness of fit and show the results on
a graph.

The expected numbers are left in L3, and you should
**verify that they meet the requirements for a χ² test**:
none of them <1, and no more than 20% of them <5.

**Example 1—shifting populations**
from Sullivan 2011 [full citation at http://BrownMath.com/swt/sources.htm#so_Sullivan2011], Example 3, pages 562–563.

Region | 2000 Census | current sample |
---|---|---|

Northeast | 19.0% | 274 |

Midwest | 22.9% | 303 |

South | 35.6% | 564 |

West | 22.5% | 359 |

“An urban economist wonders if the distribution of residents in the United States is different today than it was in 2000. In 2000,” the proportions were as shown in the table. “The economist randomly selects 1500 households and obtains the frequency distribution shown. ... Conduct the appropriate test to determine if the distribution of residents in the United States is different today from the distribution in 2000, using the α = 0.05 level of significance.”

**Solution**: H_{0} is that the model is still good, and
H_{1} is that the distribution of US population has changed.
The model is the percentages shown, and you
are determining whether the current sample is enough different from
the model for you to conclude that the model is no longer correct.

Put the percentages in L1; there’s no need to convert them to decimals but if you do then you must convert all of them. The current sample is the Observed numbers, and they go in L2.

**Caution**: The Observed numbers must always be actual
counts; never convert them to percentages.

**Caution**: Some problems will give you total
observations, but you never enter the totals in your calculator.

Run the `MATH200A`

program, select `6:GoF test`

, and press [`9`

] to confirm
that you have data in the correct lists.

The results screens are shown at right. None of the expected values are below 5, so that requirement for the goodness-of-fit test is met. There are three degrees of freedom (four categories minus one). The χ² test statistic is 8.25, and the p-value is 0.0410.

**Conclusion**: Since p<α, you reject H_{0}
and accept H_{1}. At the 0.05 level of significance, you conclude
that the regional distribution of US residents is different today
from what it was in 2000.

The screen gives a summary, but there’s useful information in L3 and L4.

L3 contains the **expected numbers**. Always check them
after the computation to make sure that none of them are below 5.
(Some authors use a looser requirement: no E’s below 1 and no more
than 20% of them below 5.) Here, all are well above 5.

L4 shows how far off the Observed numbers are, taking
into account the weights of Expected numbers in the model.
Or you can say that L4 shows each category’s
**contribution to χ²**. You can see that the most important
deviation is in the second category (4.78), and the least important is in the
first category (0.42).
The total of L4 is the
**χ² test statistic**, which you’ve already seen has
a value of 8.25 for this test.

As a rule of thumb, a value above 4 in a row of L4 means that that row’s observations are notably different from the model. Here the value L4(2) is the only one above 4. Look at observed and expected in the same row. L2(2) is 303 and L3(2) is 343.5: observed is less than expected. Row 2 is the Midwest, so the Midwest is significantly under what you would expect from the model.

**Example 2—equal frequencies**
from Sullivan 2011 [full citation at http://BrownMath.com/swt/sources.htm#so_Sullivan2011],
Example 5, pages 564–565.

Sample of Birth Records | |
---|---|

Day of Week | Freq- uency |

Sunday | 57 |

Monday | 78 |

Tuesday | 74 |

Wednesday | 76 |

Thursday | 71 |

Friday | 81 |

Saturday | 63 |

“An obstetrician wants to know whether or not the proportion of children born each day of the week is the same. She randomly selects 500 birth records and obtains the data shown.” Clearly the frequencies in the sample vary from day to day, but do they vary enough that you can say babies in general are born with different frequencies on different days, at the 0.05 level of significance?

**Solution**: H_{0} is that babies are born with equal
frequency on all days of the week, and H_{1} is that they are
not. You’re not given a numerical model because “equal
frequencies” means that all model numbers are the same. Since
there are seven categories, your model is seven 1’s in L2. The
Observed numbers go in L2, as usual.

The results are shown at right. Here no E’s are < 5; in
fact they are all about 71. df = 6 because there
are seven categories. The χ² test statistic is 6.18, and the
p-value is 0.4029. That is greater than α, and so you fail to
reject H_{0}.

Writing non-conclusions is problematic when p>α in a χ² test. Strictly speaking, the conclusion should be in neutral language as usual: you can’t determine from the data whether babies are born with equal frequency on all the days of the week or not. But traditionally, the conclusion is often stated in some words equivalent to “the data do not rule out the model”.

This is the **scientific method**. If the
experiment is repeated multiple times and H_{0} is never rejected, we
begin to have more and more confidence that H_{0} is actually true.
We can’t accept H_{0} from a single experiment, but the more
times it’s not rejected, the more we believe that it may never
be rejected.

This section of the program includes the calculations of the
TI-83’s `χ²-Test`

and provides additional
information.

**Example 2—side effects of drugs**,
adapted from Sullivan 2011 [full citation at http://BrownMath.com/swt/sources.htm#so_Sullivan2011], pages 579–581:
In clinical trials, patients with high cholesterol were randomly
assigned to three groups. One group was given Zocor, one group was
given a placebo, and one group was
given cholestyramine (a generic drug). Below are the numbers of those
in each group who did and did not experience abdominal pain. At the
0.01 significance level, is a person equally likely to have abdominal
pain on all three regimes?

Zocor | Placebo | Cholestyramine | |
---|---|---|---|

Abdominal pain | 51 | 5 | 16 |

No abdominal pain | 1532 | 152 | 163 |

**Solution**: This is a test of independence.
H_{0} is “abdominal pain is equally likely with all three
treatments”, and H_{1} is “abdominal pain is not equally
likely with all three treatments”—not just in this sample
of patients, but in cholesterol patients generally.

Press [`2nd`

`x`

^{-1}*makes* `MATRIX`

] (on the classic TI-83, press
[`MATRX`

]), then [`◄`

] [`ENTER`

] to edit
matrix A. You have 2 rows and 3 columns. (In this problem, row and
column totals are not shown, but even if they were you would ignore
them.)

Run the `MATH200A`

program and select `7:Two-way table`

. Matrix A is the only input;
the program will immediately start computing and in a few seconds will
display the results.

What is this screen telling you?
First, **requirements** for the χ² test are checked.
As you know, all expected values must be at least 5.
(Some authors give a looser requirement: no more than 20% of E’s
below 5, and no E’s below 1. When there are expected values
below 5, MATH200A also shows the number of
expected values that are below 1.)
In this case no E’s are below 5, so you’re fine.
The expected values themselves can be seen in matrix B.

There are two degrees of freedom
(*df* = rows−1 times columns−1).
The test statistic is
χ² = 14.71, quite large for *df*=2.
The **p-value** is therefore
quite low, 6.4×10^{-4} or 0.0006.
**Conclusion**: Whether
you take Zocor, nothing, or cholestyramine does make a difference to
your likelihood of experiencing abdominal pain.

The last two line tells you that the program has put
additional information in matrix C, so let’s have a look there.
Press [`2nd`

`x`

^{-1}*makes* `MATRIX`

] (on the classic TI-83, press
[`MATRX`

]), then [`◄`

] [`3`

] to view
matrix C. Unfortunately you can see only three columns at a time on
your calculator, but I’ve pasted two screens together here for
convenience:

The program dimensions matrix C two rows and two columns larger than input matrix A. This is what matrix C tells you:

- The upper left rectangle, of the same size as matrix A, is the
contribution from each cell to the total χ² for the table.
These χ² contributions are also
known as the
**squares of standardized residuals**, and as a rule of thumb a value above 4 is probably significant.Since matrix A was 2×3, look at the first 2 rows and 3 columns of C. All the χ² contributions are quite modest, except for row 1, column 3. Referring back to the original data table, you see that that’s the number of people in the cholestyramine group who experienced abdominal pain. This

*suggests*that abdominal pain with cholestyramine is either more likely or less likely than with either nothing or Zocor, not just in the sample where it certainly was different, but in people in general. But be careful! Your significance level relates to the whole χ² test, and**the p-value apples to the table as a whole, not to any one cell.** - The next row and column are the totals from the original
matrix. Since the original matrix was 2×3, its totals are in
row 3 and column 4 of matrix C.
Look at the column totals (row 3 of this matrix). 1583 patients were in the Zocor group, 157 in the placebo group, and 179 in the cholestyramine group. The row totals (column 4), show that in the entire sample, 72 patients had abdominal pain and 1847 did not. And whichever way you add, the total number of patients in the study was 1919.

- Finally, the last row and last column of matrix C are the overall row and
column percentages. (The percentages add to 100 in either direction,
apart from rounding errors.) If H
_{0}is true, you would expect to see these same percentages in each row and each column, not just in the totals.Your H

_{0}was that all three regimes are equally likely to lead to abdominal pain. In the sample overall, 3.752% (about 4%) had abdominal pain, and 96.248% (about 96%) did not. If H_{0}is true and the likelihood of abdominal pain is the same for all three regimes, then you would expect 4% of each group to experience abdominal pain: 4% of the Zocor group, 4% of the placebo group, and 4% of the cholestyramine group. But referring back to the original data, you see that 51/1583≈3% of the Zocor group, 5/157≈3% of the placebo group, and 16/179≈9% of the cholestyramine group experienced abdominal pain. This is why the χ² contribution for cholestyramine is so large in this study.

**Example 2—marital status and happiness**,
adapted from Sullivan 2011 [full citation at http://BrownMath.com/swt/sources.htm#so_Sullivan2011], pages 575–576:
In the 2006 General Social Survey, a random sample were asked their
marital status and level of happiness, with the results below. Can
you say, at the 0.05 significance level, that happiness level and
marital status are dependent?

Married | Widowed | Divorced/ Separated | Never Married | Totals | |
---|---|---|---|---|---|

Very happy | 600 | 63 | 112 | 144 | 919 |

Fairly happy | 720 | 142 | 355 | 459 | 1676 |

Not too happy | 93 | 51 | 119 | 127 | 390 |

Totals | 1413 | 256 | 586 | 730 | 2985 |

**Solution**: H_{0} is “marital status and happiness level
are independent”, and H_{1} is that they are dependent (or associated,
or not independent).
Enter the numbers in 3×4 matrix A, run the `MATH200A`

program, and select
`7:Two-way table`

.

**Careful!** Even if you are given row or column
totals, you are concerned only with the detailed numbers. You have
three happiness levels and four marital statuses, so this is a
3×4 matrix.

Results: No expected values are below 5.
The χ² statistic of
224.12 is enormous, and the p-value of 1.4×10^{-45} is
correspondingly tiny, so you conclude
that happiness level and marital status are indeed dependent.

Matrix C looks like this (again, after pasting two screens together):

Since matrix A is 3×4, the first 3 rows and 4 columns of matrix C are the χ² contributions, or squares of standardized residuals. Many of them are quite large, showing that “happiness is independent of marital status” fails badly as a model.

The row and column totals are next, but in this case you already know the totals because they were given in the problem.

Finally, look at the row and column percentages. About 31%
of the sample described themselves as very happy, 56% as fairly happy,
and 13% as not too happy. If H_{0} is true, you would expect 31% of
married people to be very happy, 56% of married people to be fairly
happy, and 13% of married people to be not too happy; and you would
expect those same percentages for widowed people, for
divorced or separated people, and for never-married people. In fact,
the percentages are quite different, as you can see if you calculate a
few of them.

The column percentages can be interpreted in the same way. About
47% of the sample was married, so if H_{0} is true you would expect
47% of very happy people, 47% of fairly happy people, and 47% of
not-too-happy people to be married. (The same statement applies in
the other columns, with their percentages.) Again, this is obviously
not the case if you look at the original observed data.

**30 Mar 2016**: Add the formula
for critical value in the normality check, for users of the
“TI-83 classic”.

**27 Dec 2015**:

- Program version 7.2: sense the calculator’s screen resolution, using a technique I learned in the forums at Cemetech, and adjust the splash screen and normality check screen layouts accordingly. Numerical results from the normality check are now shown on the graph screen again.
- Simplify instructions for downloading the program.
- Show input screen for the normality check.
- Shrink most screen shots by 25%.

**13 Dec 2015**: Program version 7.1: update Web address,
show compatibility on splash screen, and shorten menu header.

**17 Oct 2015**: Add TI Connect CE in
Getting the Program.

**16 May 2015**: Mention binomial probability distribution
in the list of histograms that the
program can create.

(intervening changes suppressed)

**Dec 2008**: Program version 1.

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