# Hints for Statistics Using a TI-83

Procedures:

(To find P(X>=x), note that P(X>=x) = 1 - P(X<=x) for any continuous distribution. )
(To find P(X>=x), note that P(X>=x) = 1 - P(X<=x-1). )

11. One-sample and two-sample Z tests: Given one or two normal samples with known population standard deviations, test H0:muX=mu0 (one sample) or else H0:muX=muY (two samples).

12. One-sample and two-sample T tests: Given one or two normal samples with UNKNOWN population standard deviations, test H0:muX=mu0 (one sample) or else H0:muX=muY (two samples).

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RESETTING the TI-83:   Press (2nd)+ (for MEM) and then enter EITHER 4:ClrAllLists or 5:Reset depending on the model of the TI-83. Follow the instructions. To do this quickly, enter EITHER (2nd)+ (for MEM) 4:ClrAllLists ENTER (TI-83 Plus etc) OR (2nd)+ (MEM) 5 then 1 then 2 (TI-83 etc).

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To do this, you must first enter a list of numbers in one of the TI-83's spreadsheet-like list registers. First,

(i) If it is not already on, press the ON button
(ii) Press the STAT button,
(iii) Make sure that EDIT is highlighted in the STAT window (you may have to use the cursor buttons). Press 1 for 1:Edit,
(iv) Move the cursor to the column heading of L1 (for List 1). If there are numbers in L1 already, or if you cannot find list L1, RESET the TI-83 as above. This will clear all lists. Start over.
(v) Enter your numbers in sequence using the number keys and either the down-arrow or the Enter key,
(vi) When you are done, press STAT again and select the CALC display in the STAT window,
(vii) Press 1 for 1:1-Var Stats (this means single-sample statistics),
(viii) When a window with 1-Var Stats appears, either press ENTER for list L1 (if that is the only list in use) or press (2nd)1 (for List 1 if you have other lists.
A number of sample statistics will now appear. Sx is the sample standard deviation. The similar but slightly smaller number (sigma)x is the population standard deviation for the sample. The population standard deviation is only appropriate if you are absolutely sure that the sample mean Xbar is exactly the same as the underlying population mean, which rarely happens in practice.
As a check, the number of elements in the list (``n'') is also listed. Make sure that this is what you expected. If you scroll down in the window, you will see the sample median and the quartiles Q1 and Q3.

For example, suppose that you know Xbar=26.13, Sx=4.815, and n=10 and want to find the confidence interval (Xbar-1.95996Sx/root(10), Xbar+1.95996Sx/root(10)) without entering Xbar, 1.95996, and Sx twice. First define a variable W for the confidence-interval half-width HWID by entering 1.95996*4.815/root(10 (with X for *, Div for /, and (2nd)x^2 for root( ) then enter STO W (ALPHA(-) for W). This displays HWID and also stores it in the variable W. Any of the alphabetical characters A-Z can be used as a TI-83 variable name. Enter W (ALPHA(-)) ENTER to display W.

Next, enter 26.13 STO M (ALPHA(Div)) to store 16.15 in M. Finally, enter M-W (ALPHA(Div)-ALPHA(-)) to display the lower confidence-interval bound and M+W (ALPHA(Div)+ALPHA(+)) to display the upper CI bound. You can also define lists and do vector operations on Lists. (See the TI-83 manual.)

(i) If it is not already on, press the ON button,
(ii) Press (2nd)VARS for DISTR and then 2 for `normalcdf(`
(iii) The syntax to calculate P(A<=X<=B) when X is a normal distribution with mean mu and standard deviation sigma is `normalcdf(a,b,mu,sigma)`. That is, after `normalcdf(`, enter `A` then COMMA (the key with a comma on it) then `B` then COMMA then `mu` then COMMA then `sigma` then ENTER. You can enter `)` (the right-parenthesis button) before you press ENTER, but it is not necessary. The probability will appear.
For example, to calculate P(20<=X<=22.7) when mu=20 and sigma=1.8, enter 20 then COMMA then 22.7 then COMMA then 20 then COMMA then 1.8 then `)` (right-parenthesis) then ENTER. The number 0.43319.... should appear.
(iv) To calculate P(A<=Z<=B) when Z is a standard normal distribution (that is, mean 0 and standard deviation 1), follow the steps in (iii) above but leave out the mu and sigma and the COMMAs before them. For example, to calculate P(0<=Z<=1.50), enter 0 then COMMA then 1.5 then ENTER. The answer should be the same as in (iii).
NOTE: To calculate P(X<=x) when X is a normal distribution with mean mu and standard deviation sigma, follow the steps in (iii) with a large negative number (for example, `-1E99`) in place of the lower bound. Here
(a) `-` is the ``unary'' minus that appears as `(-)` on the TI-83 keyboard and NOT the ``binary'' minus that appears on the right-most column of buttons and as used for expressions like 13-17, and
(b) `E` is entered by entering 2nd COMMA for EE.
Here `-1E99` is scientific notation for -1 followed by 99 zeroes and is meant to represent ``-infinity''. Similarly, calculate P(X>=x) by entering `1E99` as the upper bound. For example, to calculate P(X<=22.7) for a normal distribution with mean 20 and standard deviation 1.8,  get to a screen with `normalcdf(` as in step (ii). Enter `-1E99` as above then COMMA then 22.7 then COMMA then 20 then COMMA then 1.8 then `)` (right-parenthesis) then ENTER. The answer 0.93319.... should appear.

(i) If it is not already on, press the ON button,
(ii) Press EITHER (2nd)0 for CATALOG or ELSE (2nd)VARS for DISTR .
(iii) For P(X<=x), scroll down until the cursor is next to `binomcdf(`  and press ENTER . Select `binompdf(`  for P(X=x). As a shortcut, if you entered (2nd)VARS, enter A (ALPHA(MATH)) for `A:binomcdf(` or 0 for `0:binompdf(`.
(iv) The basic syntax for the two commands in (iii) is `binomcdf(numtrials,p,x)` and `binompdf(numtrials,p,x)`. For example, assuming that you want P(X<=x), wait for a window with `binomcdf(`  to appear. Enter `n` then COMMA (that is, press the button with a comma on it) then `p` then COMMA then `x` then ) (the right-parenthesis button) then ENTER. The probability P(X<=x) should then appear. For example, If n=50, p=0.055, and x=3, enter 50 then COMMA then 0.055 then COMMA then 3 then ) then ENTER. The number 0.70469... will then appear.
(v) To find P(X>=x), note that P(X>=x) = 1 - P(X<=x-1) for any integer-valued random variable X.  You can calculate probabilities like P(a<=X<=b) by subtraction.

You can do some fancier things as well:
(vi) To get all values of P(X<=x) at once, use the syntax `binomcdf(numtrials,p)`. That is, after you get to the window with `binomcdf(`, enter `n` then COMMA then `p` then ) (right parenthesis) then ENTER, without the variable x. A list of the values of P(X<=x) will appear, most of which will be outside of the calculator window. To view them in a list, enter `STO(arrow)` then (2nd)1 (for List 1) then ENTER, then enter `STAT` then 1 for `1:Edit`. The values of P(X<=x) will be displayed in a list.
(vii) To calculate P(X=x) for n=50, p=0.055, and all values of x between 0 and 7 inclusively and display the probabilities in list L2, first enter the values 0,1,2,3,4,5,6,7 in list L1. (See the instructions above for sample mean and standard deviation.) Enter (2nd)MODE for QUIT so that the calculator will know that you have stopped entering values in list L1. (Otherwise, the calculator will assume that any further operations have to do with the next entry in L1.) Enter (2nd)0 for CATALOG, scroll down to `binompdf(`, and press ENTER. When the window with `binompdf(` appears, enter 50 then COMMA then 0.055 then COMMA then (2nd)1 for list L1 then ) (the right-parenthesis button to close the `binompdf(` function) then `STO(arrow)` then (2nd)2 for list 2 then ENTER. If you press STAT and then 1 for 1:Edit, then the eight probabilities P(X=x) for x=0,1,2,3,4,5,6,7 will be in list L2 .

NOTE: Olders TI-83s may crash if you try to calculate a binomial cumulative probability with n=1000 . Use the normal approximation to the binomial for values of n that are this large.

(i) If it is not already on, press the ON button,
(ii) Press EITHER (2nd)0 for CATALOG or ELSE (2nd)VARS for DISTR .
(iii) For P(X<=x), scroll down until the cursor is next to `poissoncdf(`  and press ENTER . Select `poissonpdf(`  for P(X=x).
(iv) The basic syntax for the two commands in (iii) is `poissoncdf(mu,x)` and `poissonpdf(mu,x)`. For example, assuming that you want P(X<=x), wait for a window with `poissoncdf(`  to appear. Enter the value of `mu` then COMMA (that is, press the button with a comma on it) then `x` then ) (the right-parenthesis button) then ENTER. The probability P(X<=x) should then appear.
(v) To find P(X>=x), note that P(X>=x) = 1 - P(X<=x-1) for any integer-valued random variable X.  You can calculate probabilities like P(a<=X<=b) by subtraction.

(i) If it not already on, press the ON button,
(ii) Press (2nd)VARS for DISTR,
(iii) Press 5 for `5:tcdf(,`   7 for `7:X2cdf(`   or 9 for `9:Fcdf(`,
(iv) A screen will appear in which you can enter the parameters for the distribution value.
The syntax for `tcdf` and `X2cdf` is `(Function)(Lower,Upper,df)`.
For example, to find P(X>=1.645) where X has a Student's t-distribution with 11 degrees of freedom, enter (in order) 1.645 then COMMA then 1 2ndCOMMA (for EE) 99 (for 1E99) then COMMA then 11 then )(right parenthesis) then ENTER. The number 0.0641075923 should appear.
The syntax for `Fcdf` is `Fcdf(Lower,Upper,numdf,denomdf)`.
Chi-square and F distributions are restricted to nonnegative values, so that you can enter 0 (zero) in place of -1E99 for the lower bound for `7:X2cdf(` and `9:Fcdf(`.
Note: `tcdf(` accepts fractional numbers of degrees of freedom, so that it can be used to get an exact value for Satterthwaite's two-sample T-test. If you enter fractional numbers of degrees of freedom for `Fcdf(` on current TI-83s, you appear to get error messages.

(i) If it is not already on, press the ON button,
(ii) Press (2nd)VARS for DISTR,
(iii) Press 3 for `3:invNorm(`.
(iv) A screen will appear in which you can enter parameters for the inverse normal distribution. The syntax for `invNorm(` is `invNorm(pp,mu,sigma ENTER` is or `invNorm(pp ENTER` for a standard normal quantile (mu=0 and sigma=1).
For example, to find x such that P(X<=x)=p when X is a normal distribution with mean mu and standard deviation sigma, enter `p` then COMMA (the key with a comma on it) then `mu` then COMMA then `sigma` then `)` (the right-parenthesis button) then ENTER. The probability will appear. For example, to find x such that P(X<=x)=0.666 when mu=20 and sigma=1.8, enter 0.666 then COMMA then 20 then COMMA then 1.8 then `)` (right-parenthesis) then ENTER. The number 20.777... should appear.
(iv) To find x such that P(Z<=x)=p when Z has a standard normal distribution (that is, mu=0 and sigma=1), follow the steps in (iii) but leave out the mu and sigma. For example, to find x such that P(Z<=x)=0.666, enter 0.666 then `)` (right-parenthesis) then ENTER. The number 0.42889... should appear.

This is harder than finding normal quantiles, and makes use of general Equation Solver routines on the TI-83 in combination with the built-in functions for the Student-t, Chi-square, or F-distribution cdfs.

For example, to find the 5% upper critical value (same as the 0.95 quantile) for a Student-t distribution with 8 degrees of freedom,

(i) If it is not already on, press the ON button,
(ii) Press MATH then 0 for 0:Solver. You should get a screen with EQUATION SOLVER at the top. If the screen is cluttered, reset the TI-83 and start over.
(iii) The cursor should be just after `eqn:0=` on the second line of the screen. Enter 0.95 then (binary) Minus (the minus key on the right-hand column of buttons), then
(iv) Enter the Student-t function `tcdf(` by entering (2nd)VARS then 5 for `5:tcdf(`. DO NOT spell out ``tcdf('' on the TI-83 keypad. You should now see `eqn:0.95-tcdf(` on the second line of the TI-83 screen.
(v) Enter -99999 (or -1E99) then COMMA then X (as (ALPHA)STO) (X will be the unknown for the TI-83 Solver) then COMMA then 8 (for 8 degrees of freedom) then ENTER,
(vi) The TI-83 Solver uses an iteration procedure to find solutions. The third line on the screen should be ``bound=(lowervalue,uppervalue)''; the starting value of the iteration is the average of lowervalue and uppervalue. You generally will not have to do anything on this screen unless (a) the iteration does not converge, which will almost never happen if you enter a single cdf function as in this case, or (b) you had entered an expression with more than one unknown variable, such as X and Y or X and A and B. In case (a), you have probably entered an out-of-bounds value in place of 0.95, which you should change. Otherwise, you can change the limits in the ``bound'' statement and have the TI-83 try again. In case (b), the TI-83 manual says that you should set fixed values at all but one variable and place the cursor at the variable that you want to solve for.
(vii) Finally, press the SOLVE (that is, ALPHA(ENTER)) key. The TI-83 may take up to 30 seconds to find the solution, which is a long time if you are looking at the TI-83 patiently waiting for your answer. You can tell that the TI-83 is still thinking since there will be a moving dot at the extreme upper right of the TI-83 screen. If the TI-83 has been successful, a number will be returned (for X) and the TI-83 will display something like `left-rt=0`, which is a sign that it has checked the answer. The returned number for X will be the desired quantile.
(viii) As another example, to find the upper F-distribution quantile F(5,10,0.01), enter (2nd)PLUS 5-1-2 (see reset above) then MATH then 0 (for 0:Solver) then 0.99 then (binary) Minus then (2nd)VARS 9 (for `Fcdf(`) then 0 then COMMA then X (that is, ALPHA(STO)) then COMMA then 5 then COMMA then 10 then ENTER. After the SOLVER screen appears, make sure that the cursor is on the X= row and enter SOLVE ((ALPHA)ENTER). After a number of seconds the number 5.636326... will appear followed by `left-rt=0`. You can check F(5,10,0.01)=5.636326... from Stat tables.

To find a confidence interval for the mean from a sample X1, X2, ... Xn, you have two choices for entering the sample data into the TI-83, either (a) enter the numbers X1,X2,...,Xn into one of the TI-83 list registers (for example, L1) or (b) enter the sample statistics Xbar, Sx, and n directly.

To find a normal (Z) confidence interval for the mean by entering the X1,X2,...,Xn explicitly,

(i) If the TI-83 is not already on, press the ON button,
(ii) Enter the numbers X1,X2,....,Xn into (for example) list L1 in the TI=83. See the sample mean and sample standard deviation section for how to do this.
(iii) Enter STAT and make sure that TESTS is highlighted by moving the cursor if necessary,
(iv) Enter 7 for `7:ZInterval`
(v) Since you have entered X1,X2,...Xn explicitly, make sure that DATA is highlighted, moving the cursor if necessary. Since this is a Z-interval, you need to know sigma explicitly. Enter the value of sigma after `sigma=`, make sure that `List:` has the correct list register (for example, L1) and that `Freq=1`. Enter the size of the confidence interval (that is, 0.95 for a 95% confidence interval). Highlight `Calculate` and press ENTER.
The confidence interval will appear, along with the values of Xvar, Sx, and n.

To find a normal (Z) confidence interval for the mean by entering sigma, Xbar, and n explicitly,

(i) If the TI-83 is not already on, press the ON button,
(ii) Enter STAT and make sure that TESTS is highlighted by moving the cursor if necessary,
(iii) Enter 7 for `7:ZInterval`
(iv) Make sure that STATS is highlighted, moving the cursor if necessary and press ENTER. Enter values for sigma, Xbar, n, and the size of the confidence interval (that is, 0.95 for a 95% confidence interval). Highlight `Calculate` and press ENTER.
The confidence interval will appear.

To find a confidence interval for the mean from a sample X1, X2, ... Xn, you have two choices for entering the sample data into the TI-83, either (a) enter the numbers X1,X2,...,Xn into one of the TI-83 list registers (for example, L1) or (b) enter the sample statistics Xbar, Sx, and n directly.

To find a Student-t confidence interval for the mean by entering the X1,X2,...,Xn explicitly,

(i) If the TI-83 is not already on, press the ON button,
(ii) Enter the numbers X1,X2,....,Xn into (for example) list L1 in the TI=83. See the sample mean and sample standard deviation section for how to do this.
(iii) Enter STAT and make sure that TESTS is highlighted by moving the cursor if necessary,
(iv) Enter 8 for `8:TInterval`
(v) Since you have entered X1,X2,...Xn explicitly, make sure that DATA is highlighted, moving the cursor if necessary. Make sure that `List:` has the correct list register (for example, L1) and that `Freq=1`. Enter the size of the confidence interval (that is, 0.95 for a 95% confidence interval). Highlight `Calculate` and press ENTER.
The confidence interval will appear, along with the values of Xvar, Sx, and n.

To find a Student-t confidence interval for the mean by entering Xbar, Sx, and n explicitly,

(i) If the TI-83 is not already on, press the ON button,
(ii) Enter STAT and make sure that TESTS is highlighted by moving the cursor if necessary,
(iii) Enter 8 for `8:TInterval`
(iv) Make sure that STATS is highlighted, moving the cursor if necessary and press ENTER. Enter values for Xbar, Sx, n, and the size of the confidence interval (that is, 0.95 for a 95% confidence interval). Highlight `Calculate` and press ENTER.
The confidence interval will appear.

(i) If the TI-83 is not already on, press the ON button,
(ii) Enter STAT and make sure that TESTS is highlighted by moving the cursor if necessary,
(iii) Enter A (ALPHA(MATH)) for `A:1PropZint`
(iv) On the screen that appears, fill in X: (the number of ``successes''), N: (the number of trials), and set the size of the confidence interval (for example, 0.95). Scroll down to CALCULATE and press ENTER. The normal-theory interval should appear.
Given one or two normal samples with KNOWN population standard deviations, test H0:muX=mu0 (one sample) or else H0:muX=muY (two samples).

(i) If the TI-83 is not already on, press the ON button,
(ii) Enter STAT, then move the cursor to highlight TESTS, then enter either 1 for a (one-sample) Z test or 3 for a 2-sample Z test.
(iii) Highlight either Data (if you have already entered your data into list L1 for one sample or lists L1 and L2 for two samples) or Stats (if you want to enter summary statistics instead). See 1. Calculating sample mean and sample standard deviations for discussion about using Lists with a TI-83.
(iv) If you want to enter summary statistics for your data, make sure that Stats is highlighted. Press Enter to make sure that your choice is fixed. Fill in the appropriate entries.
(v) Leave any Freq entries at 1.  Highlite the correct choice for a two-sided test, lower-tailed one-sided test, or upper-tailed one-sided test and press Enter to make sure that the choice is set. Finally, highlite Calculate and press Enter.
If your data is in terms of summary statistics (Xbar or Xbar and Ybar), make sure that Stats is highlighted instead of Data and continue in the same way.
Given one or two normal samples with UNKNOWN population standard deviations, test H0:muX=mu0 (one sample) or else H0:muX=muY (two samples).

After you have turned on the TI-83 and possibly reset or cleared it, enter STAT, then TESTS and then either 2: for a (one-sample) T test or 4: for a 2-sample T test. In either case, the first step will be to either highlite either Data or Stats.

If your data is in terms of the samples themselves, make sure that Data is highlighted. Press Enter to make sure that your choice of Data or Stats is fixed. Fill in mu0 (for the one-sample test) and the lists (L1 or L1 and L2) that you have entered your data in. (See 1. Calculating sample mean and sample standard deviations for discussion about using Lists with a TI-83.) Leave any Freq entries at 1.  Choose a two-sided test, lower-tailed one-sided test, or upper-tailed one-sided test and press Enter to make sure that the choice is set. In the 2-sample case, select Pooled=Yes for the Classical 2-sample t-test or Pooled=No for the Satterthwaite Test. Finally, highlite Calculate and press Enter.
If your data is in terms of summary statistics (either Xbar and Sx or else Xbar, Sx, Ybar, and Sy), highlite Stats instead of Data and continue in the same way. Enter the sample standard deviation or standard deviations rather than the sample variance or variances.

13. Given paired data (X1,Y1),  (X2,Y2),  (X3,Y2),  ...,  (Xn,Yn),  find

(a) The Pearson correlation coefficient between Xi and Yi and
(b) The coefficients of the linear regression Y = aX+b     Go to top of this page

(i) If the TI-83 is not already on, press the ON button,
(ii) Enter STAT, then 1 for 1:EDIT, then enter the Xi into list L1 and Yi into list L2. If there is data already in lists L1 and L2, it may be convenient to first reset or clear the TI-83 to clear the lists.
(iii) If you want the TI-83 to calculate the Pearson correlation coefficient r (which is the same as the Model R2 = r2 for the linear regression), you must turn ``Diagnostics'' ON by entering `2nd 0` (for CATALOG), space down to `DiagnosticOn`, press ENTER, and then ENTER again if you see `DiagnosticOn` on a different screen,
(iv) Enter STAT then highlight CALC then enter 4 for `4:LinReg(ax+b)`.
When a new screen appears, enter `2nd 1` for list L1, then COMMA (the comma key), then `2nd 2` for list L2, then ENTER. After a few seconds the coefficients a,b of the regression `Y=aX+b` will appear. If you entered `DiagnosticOn` in step (iii), then r2 and r will also appear.

After you have turned on the TI-83 and possibly reset or cleared it,

(a) Enter STAT, then 1:EDIT, then enter the d samples in your data into d lists L1 through Ld. (For example, L1 through L4 if there are four ``treatments''.)
(b) Enter STAT then highlight TESTS then enter F (ALPHA(COS)) for `F:ANOVA(`. If d=4 for four treatments, the syntax is `ANOVA(L1,L2,L3,L4)`. You will have to scroll through several screens for the entire output.