STAT CALCULATIONS (TI-83)

$tiger$

~IMPORTANT NOTE: There are times when error messages occur & you just can't seem to find out why...since everything you're doing is correct. This has happen to me on several occasions & can be very frustrating & time-consuming. So, if you find yourself in this situation, reset your calculator to the factory defaults & try again. However, you will probably lose saved programs..

~To do this, use: 2nd mem(+), (7) Reset, enter, (2) Defaults, enter, Reset, enter.

~Note: Our textbook has complete descriptions of the procedures used for the TI-83/84 Plus throughout the sections of the text. However, I'll explain all of the menus used in my course in this article.

~You'll need the TI-83 or equivalent for my lessons, quizzes, & exams.
You need to bring it to every class meeting. If you don't & there is a quiz or exam, you will not be allowed to borrow your neighbors & will have to know all formulas necessary by memory. The TI-83 is a requirement for this course & if you don't have one, the probability of passing this course is very small...please don't place yourself in this very bad situation.

~Note: I will allow you to use your calculator for all computations. You must practice & learn how. If not, you will not be able to finish questions on quizzes & exams. See me, if you are having trouble pressing the right buttons.

~Note: The midrange (Extreme Value Mean), range, mode are very easy to get without a calculator.

~Generating k random integers from L (low number) to Q (high number): Use MATH, PRB, (5)randInt, enter, insert L, comma, insert Q, comma, insert k, enter.

~Note: this is great for selecting random samples from populations (you must assign numbers to data pts within the population before selection)

~Entering data in lists:
Clear lists by STAT,4,ENTER,2nd L1,2nd L2, & so on.
Then use Stat, Edit, Enter, Data value, Enter, Data value, Enter, repeat until all data pts are listed in L1, then, use Stat, Calc, 1-Var Stats, Enter, 2nd, L1, Enter, results will be displayed. (all data values here have a frequency of 1) (if the frequencies are other than 1, you must enter them in L2)
Results displayed are Sample mean, Sum of data values, Sum of squares of the data values, Sample standard deviation, Population standard deviation, n = the number of data pts (size of the sample), also, moving curser down gives Q1, Med(Q2), Q3.

For HISTOGRAMS, follow the following procedure:

1) First, clear your lists: press STAT, go to 4, Enter, 2nd L1, 2nd L2, Enter (see done).

2) Now, press STAT, stay at 1, Enter (should see empty lists). Move cursor to left list L1 & enter the midpoints of each interval (this is the horizontal axis) (X axis). You can do it by (left value+right value)/ 2 Enter for each one

3) Once all those are entered, move cursor to L2 & enter each frequency.

4) Now, press 2nd STAT plot, stay at 1, Enter, make sure ON is blinking, Enter, move cursor to third on the top (displays Histogram plot), Enter, move to L1, enter, move to L2, enter, then hit Zoom 9 (see note below) & SMILE!.....

Note: When using Zoom 9, Xsci (class width) must be adjusted.

Adjusting the viewing window manually: Press window. For Xmin use the smallest X (data value) & for Xmax, the largest. For Xsci use the class width (width of each interval). For Ymin, use 0, for Ymax, use a little more the the largest frequency.

For BOX-PLOTS, follow the following procedure:
~the smallest data value (min), Q1, Med(Q2), Q3, & the largest data value (max) are known as the 5 number summary & are used to construct box-plots. Once the data has been entered into your calculator, the TI-83 will do this for you automatically by pressing 2nd STAT plot (far top left),(1), enter, on, move curser to center bottom chart (regular type of boxplot), enter list where data is stored (if not there already), enter Freq: 1 (if not there already), Zoom 9 (sets up viewing window).
~If you choose the first type of boxplot in row 2 (modified), you will see any outliers as pts on you boxplot. Be careful, outliers close to each other may appear as one pt.
~Note: More than one boxplot may be shown if you use more that one plot & the data has been entered in different lists.
This is a popular way of comparing two data sets.

~Note: The formula for the sample standard deviation(Sx) is different from the one that gives or estimates the population standard deviation (lower case sigma x). That is why we have two different answers here.
In many cases, the population standard deviation is not known & we use the sample standard deviation to estimate the population standard deviation. The larger the sample size, the better the estimate. In the formula, it has been found, by dividing by (n-1) instead of n gives a better estimate of the population standard deviation when it is not known.

~STATS FROM FREQUENCY DISTRIBUTIONS:
Clear lists L1 & L2 by STAT, move curser to 4 , Enter, 2nd L1, comma, 2nd L2, Enter. Now use STAT, Edit, Enter to enter the data value or midpt of the interval (compute the midpt of each class by taking the average of the class limits) (make sure you use parentheses properly) (i.e., (a+b)/2). Enter for each one in L1. Then move curser to the right (L2) & enter each frequency. When done, click STAT, CALC, 1-VarStats, 2nd L1, comma, 2nd L2, Enter. The results will be displayed.

~Note: To get the variance, just square the standard deviation. Just copy Sx in your display (do not round it off), then press the X square botton.

~STATS FROM PROBABILITY DISTRIBUTIONS:
Same as frequency distributions but enter the probabilities of each outcome in L2.

~Factorials:
The symbol for factorial is !. So, 5!=(5)(4)(3)(2)(1), & 8!=(8)(7)(6)(5)(4)(3)(2)(1), & so on.
Just insert the number n, MATH, PRB, 4, ENTER

~Permutations:
An Arrangement of n different objects taking them r at a time.
Notation: nPr = n!/(n-r)!, nPn = n!/0!=n!, just remember to enter the n first, then MATH, PRB, & find it.

~Combinations:
These deal with selection (what is there) NOT arrangement.
Notation: nCr =( nPr)/n! = n!/[r!(n-r)!], nCn=n!/[n!o!]=1, nCo =1, by definition.
Look for it in the same place, but remember, enter n before you press that item.

~Multipying and dividing combinations:
i.e., for (nCa)(nCb) or (nCa)/(nCb)
Do: insert n, MATH, PRB, menu(3), enter, a, enter, press multiply or divide,insert n, enter, MATH, PRB, menu(3), enter b, enter.

~Note: Pascal's triangle (lesson #9) can be used to get all combinations without a calculator. Here's how it works.

PASCAL'S TRIANGLE

1
1    1
1    2    1
1    3    3    1
1    4    6    4    1
1    5    10 10    5    1

& SO ON....

~To get a number, just add the two numbers directly above, except for the 1's on the ends. Reading a row, gives you the combinations of the 2nd number (left to right), taken 0, 1, 2, 3, ... at a time.

~For example,
in the above triangle, the 3rd number in the bottom row formed, is 10 (from the left). This will equal 5C2 . It turns out the 5C3 also = 10.

However, if n is large, the triangle will be very wide & this might be time consuming...much easier to use the TI-83. (see link on Pascal's Triangle)

~BINOMIAL PROBABILITIES:
a) For a single probability (fixed number of trials), use 2nd VARS, go down to 0 on the menu to binompdf( & enter n, p (probability of a success), r (number of successes), enter
b) For cumulative probabilities (sum of so many successes starting at 0), use 2nd VARS, go to A on the menu to binomcdf( & enter as in a)
Note: This button is a must to use for problems dealing with "at most" a given number of successes and for problems dealing with "at least" a given number of successes, however, for the latter, the backdoor appoach must be used to convert P(at least r successes) to
1 - p(at most r-1 successes).

~Note: We can find probabilities (areas) using the standard normal curve falling to the right, left, or between two given scores. You have the option of converting these scores to z-scores or not (see a quick way without z-scores, below). Also, we can find z scores corresponding to given probabilities, consequently, finding percentile scores & others corresponding to them.

These are done conveniently on your calculator using the 2nd VARS menus, 2 & 3.

Using z-scores: we need two z scores for finding areas (probabilities), use z=10,000 or z=-10,000 when finding areas greater or less than a given z score, respectively. (i.e., to find the probability of a score greater than z=1.5, insert z=1.5 for the left limit & z=10,000 for the right limit. For less than, insert z=-10,000 first)

Note: when going in reverse (using (3)invNorm, insert the total area to the LEFT of the z score that you want, the display will give that z-score)

~Quick way (without z-scores): For finding probability (area) between two data pts (X scores),
use: 2nd VARS, menu(2), enter, insert left X score, right X score, mean, standard deviation, enter.

~Quick way (without z-scores): for finding X score given a probability (area), use: 2nd VARS, menu(3), enter, insert area or probability, mean, standard deviation, enter. This is used for finding percentile scores in a given data set.

~Note: When working with MEANS of different size samples from a given population, probabilities & scores can be computed by using the same menus, 2nd,VARS, (2) or (3), but, when entering the standard deviation, make sure you enter the population standard deviation divided by √(n). (see link on the Central Limit Theorem under Topics of Interest).

~For an individually selected score, just enter the population standard deviation.

~These quick ways avoid finding the z scores completely, however, you need to know how to find a z-score for a data point in many types of problems. So, know both ways.

~Finding confidence intervals for estimating popultion proportions

~Note: A Confidence interval is a range or interval of values used to estimate the true value of the population proportion. (see lesson #17)

~Confidence level: This is the probability 1- alpha where alpha/2 is the area under the standard normal curve at both ends.

~The Critical value Z_alpha/2 is the positive z value separating an area of alpha/2 on the right side of the standard normal curve. (-Z_alpha/2 would correspond to the left side or tail)

for a 90% confidence level, alpha = .10, Z_alpha/2 = 1.645
for a 95% confidence level, alpha = .05, Z_alpha/2 = 1.96
for a 99% confidence level, alpha = .01, Z_alpha/2 = 2.576

~The above confidence levels are the most popular

~Note: To get these critical values, use 2nd Vars, menu 3, invNorm with .9500, .9750, and .9950 respectively.

~Ex: Find the critical value Z_alpha/2 corresponding to the 95% confidence level. Use 2nd VARS, menu 3, insert (1 - .05/2)

~Note: Since the sample proportion p cap , is typically different from the population proportion p, we call this difference the margin of error E.

~To find E, we subtract p cap from the upper limit value of the confidence interval or subtract the lower limit of the confidence interval from p cap (Assuming we can get the confidence interval).

~Here's an example illustrating how these are found using the TI-83.

~Ex: Let's take our study of the sample of 80 Marist College students , of which 65% were found to love pizza. Would this be true of all the students? Let's find the 95% confidence interval for the population p (all students). Here n=80, p cap = .65. Use STAT, Tests, down to A on the menu (PropZint), enter x= (80)(.65) = 52 (number of students who love pizza in our sample), (be careful here, since x must be an integer), n =80, .95 (C-level), calculate, enter.

~You should be viewing (.54548, .75452), or , .54548 < p < .75452 for the 95% confidence interval. Which means, we can confident, 95% of the time, that the true population proportion p lies within this range.
Now, for the margin of error E.

~Subtract p cap = .65 from the upper limit value of .75 to get E = .10 (10 % margin of error).

~If we repeated this procedure for different Confidence levels, we would get the following:
99%-----(.51264, .78736)-----E=.14
90%-----(.56229, .73771)-----E= .09
30%-----(.62945, .67955)-----E= .03

~Notice that p is the average of the end points of the confidence interval

~Notice that the larger the level of confidence, the larger the interval & E

~Estimating an unknown Population mean (population standard deviation is known)(see lesson #18)

To get a confidence interval, use STAT, Tests, Zinterval(7), enter Data or Stats, C-Level, Calculate.

~For the above, there will be an entry for the population standard deviation since it is known.

~Margin of Error (E):
To get E, subtract the lower limit of the interval displayed from the mean or subtract the mean from the upper limit .

~Note: For finite populations of size N where n>.05N, replace (Populaton S.D.)/√n by [(Population S.D.)/√n] times √[(N-n)/(N-1)] (finite population correction)[see p277].
Confidence intervals and margins of error will change.

~Estimating an unknown Population mean.
(population standard deviation is unknown)(see lesson #19)
To get a confidence interval using t-values, use STAT, Tests, Tinterval(8), enter Data or Stats, C-Level, Calculate.

~For the above, there will not be an entry for the population standard deviation since it is unknown.

~Test statistic:
This comes from our sample. The TI-83 will give it to you automatically. If you are using the standard normal curve, use STAT, TESTS, Z-test(1), fill in the information, Calculate.

If you are using t-curves (standard deviation of the population is unknown), use STAT, TESTS, T-Test, fill in the information, Calculate.

The display will give you the test statistic and its p-value underneath it.
The p-value has to do with the area under the curve you are using. It gives the area to the right of the test statistic (for a right-tailed test), the area to the left of the test statistic (for a left-tailed test) & for a two-tailed test, it will twice the area to the right or left of the test statistic (depending upon whether or not the test statistic is positive or negative).

The smaller the p-value, the more evidence we have & the more likely we are to reject Ho (depending upon what significance level we are using). A very small p-value indicates that the sample has produced a test statistic which is extreme and not inline with the null hypothesis.

In any case, IF THE P-VALUE IS LESS THAN OR EQUAL TO THE SIGNIFICANCE LEVEL, WE REJECT THE NULL HYPOTHESIS, OTHERWISE, WE DO NOT REJECT IT.

Saying it another way: the p-value is the smallest significance level at which the null hypothesis can be rejected based on the sample data.

When the p-value is less than or equal to the significance level, the test statistic will fall in the rejection region.

~Occasionally, you will have to compute a test statistic & p-value without the use of the Z-Test or T-test on your TI-83. To do this, use the formulas in the insert of your textbook for the test statistic, then find the p-value (area related to it-see insert) by the use of the TI-83 using 2nd VARS, normalcdf (input 2 z values). if dealing with the standard normal curve or use table A-3, if dealing with t-curves. However, you need to read table A-3 in reverse & probably would not find your test statistic exactly in the heart of the table. So, we often get a range of values where that p-value is located.

In most cases, that would be enough to determine whether or not the p-value is less than the significance level (for rejecting or not rejecting the null hypothesis).

The TI-83 will do all that is necessary to complete a hypothesis test. It doesn't give us critical t-values, but we don't need to find them since all we have to do is compare the p-value to the total area in the rejection region, which is the significance level (usually .05 or .01).

~Testing a claim about a proportion:
Use STAT, TESTS, 1-PropZTest(5), Enter, x (integer), n, choose H1, calculate, Enter. The display will give you the p-value. If less than or equal to the significance level, we reject Ho.

Note: the p-value is the probability that a z score is less than or equal to the test statistic.

~Testing a claim about a mean: (populaton standard deviation known)
Use the p-value method, (1) Z-Test.

~Testing a claim about a mean: (population standard deviation not known)
Note: The p value method is best here since the test statistic is calculated automatically & you do not need to get critical values.

~Note: The procedures are the same as in the previously covered tests, however, t curves are used, (2): T-Test

~Correlation: For paired data (x,y). Here y is considered the dependent variable (vertical axis). We will cover LINEAR correlation only (the best fitting straight line through our data points). If the slope of this line is positive, the correlation r is positive. If the slope is negative, so is the correlaiton r. The correlation will also be unchanged if we treat x as the dependent variable & y the independent variable. (i.e., enter y data in L1 instead of the x data). Be careful, the best fitting straight line (regression line) would change.

Fortunately, the TI-83 will do it for us under the menu: STAT, TESTS, (E)LinRegTTest.
~Note: Make sure you move the curser down far enough to view all results in the display.

~Note: To test for a significant correlation at the .05 or .01 level, look at the absolute value of r (the value of r without a negative sign) & see if it exceeds the critical value in table A-6 (see insert) for that significance level. If it does, we conclude that there is a significant correlation, otherwise, there is not sufficient evidence to support a significant correlation. Using the TI-83, we would reject the null hypothesis (r=0) (no significant correlation) when |r| exceeds the critical value in table A-6.

~Regression: Gives the equation of the straight line that best fits our paired data. As mentioned above, it does matter which of our data sets we consider independent (x) (insert in L1) & dependent (y) (insert in L2).

In the first case, we say the regression line is "y on x". In the second case, we say "x on y". These lines have different linear equations (they are inverses).

Use the same menu: STAT, TESTS, LinRegTTest(E).

~Note: If there is a significant correlation, we may use the regression line for prediction (predicting y for a given value of x). If not, the best predicted y-value would be the mean of all y-values, for ANY X-VALUE.

TI-83/84 Plus Procedure for Two Proportions

TI-83/84 Plus Procedure for Two Means

~For using various equation models for data, go to Curve Fitting

~Also see: Popular menus used in Statistics