USING THE TI-83/4 STATISTICAL PROGRAMS ***

Transcription

1 USING THE TI-83/4 STATISTICAL PROGRAMS *** STA 2023 TALLAHASSEE COMMUNITY COLLEGE

2 Harry Biske, 1999 Tallahassee Community College Tallahassee, Florida

3 TABLE OF CONTENTS Introduction 1 I. Sampling and descriptive statistics functions Page A. Generating random samples 2 Seeding the calculator Selecting a simple random sample Selecting a systematic random sample 3 B. Tables: Ungrouped or grouped data tables Entering data into Lists 4 Clearing Lists Sorting Lists Multiple Lists for grouped data C. Graphs: Generating a bar graph 4 Generating a histogram 5 D Summary measures: Ungrouped or grouped data 6 E. Generating boxplots 7 II. Linear correlation and regression functions A. Entering data 8 B. Constructing a scatterplot C. Constructing a line of regression 9 D. Drawing the line of regression E. Predicting y from the line of regression F. Constructing a prediction interval III. Probability functions A. Calculating parameters 13 B. Factorials B. Permutations 14 C. Combinations E. Binomial distribution For single values of x For cumulative values of x 15 i

4 Page IV. Normal distribution A. Probability of individual values of x or z 16 B. Probability between two values of z C. Probability between two values of x 17 D. Finding values of z if probability is known E. Finding values of x if probability is known 18 F. Assessing normality of data. V. The Central Limit Theorem and Estimation Functions A. Central Limit Theorem 19 B. Estimation functions Estimating µ x, large samples 20 Estimating µ x, small samples Estimating p x, large samples VI. Hypothesis testing functions A. Testing a hypothesis of µ x with a large sample 23 B. Testing a hypothesis of µ x with a small sample 24 C. Testing a hypothesis of p x with a large sample 25 ii

5 Tallahassee Community College H. Biske STA 2023 Revised; December, 2004 USING THE TI-83/4 STATISTICAL PROGRAMS The Texas Instruments TI-83, TI-83 Plus, TI-83 Silver or TI-84 calculator is required equipment for students taking STA It is the best calculator for statistical functions, surpassing other models and even later TI models. The TI-83 Plus and Silver models, as well as the new TI-84 has the same statistical programs as the basic TI-83, with additional memory. Beware! The TI-89 and TI-92 models do not contain the same variety of statistical applications as the TI-83, and use a different presentation format. The TI-83 statistical programs reportedly may be downloaded into the TI-89 and 92 models, using a computer-calculator linkage device; however, you would still have a different format than the TI-83/4 models. Instruction will be provided in class only on the use of the TI-83/4 to perform certain statistical operations. Instruction will not be provided for other calculators or computer statistical programs. In the following descriptions and throughout this booklet an arrowhead ( > ) is used to show a sequence of key strokes or entries. For example: 2 nd > MEM > 3 > ENTER indicates four keystrokes. Before using the TI-83/4, familiarize yourself with the keyboard: the basic arithmetic and scrolling operations in blue, the ALPHA function in green, and particularly the 2nd functions, indicated in yellow. The 2nd functions particularly are used heavily in statistical functions. Also, before using the calculator, clear any existing data that may have been stored from previous use. To clear the TI-83/4 memory enter 2nd > MEM > 3(Clear entries) > ENTER; and 2nd > MEM > 4(Clear lists) > ENTER; and 2nd > MEM > 5 (Reset) > 2 (Default). To go to the home screen for new operations, press the CLEAR button or 2nd > QUIT.

6 TI-83/4 2 I. SAMPLING AND DESCRIPTIVE STATISTICS FUNCTIONS A. Generating random samples: The TI-83/4 s random numbers generator can help you select a random sample of size n from a population of size N. 1. Seeding the random numbers generator establishes the same starting point for a group of persons using separate TI-83/4s; it will ensure that each person gets the same random numbers. It is required only for group standardization; it is not required for use of the random number generator for individual use. To set a seed, enter any number other than 0 and store it in the random number generator. Example 1: Seeding with 123 : Enter 123 > STO > MATH > PRB >1(rand) > ENTER = (resulting in) 123 rand; press >ENTER = 123. The random generator is now seeded at 123. Note: When used in a group setting, seeding may be required for each use of the random number generator, particularly if one or more persons in the group perform different or additional operations from the other members of the group after the initial seeding. It is important that each person in the group be on the same page. Performing more operations than others in the group will put you on a different page in the random setting, resulting in different random numbers. 2. To select a simple random sample of size n from a population of N: Enter MATH > PRB > 5 (randint)= randint ( enter 1 >, > N >, > n > ENTER = {list of n random integers between 1 and N }. Note the use of commas to separate 1, N and n entries; they are essential entries. If the list appears incomplete on the LCD display, i.e.,{..., scroll right with the blue (>) key. If there are duplicates in the first list of n integers, press ENTER again and a second list will appear. Continue until n distinct integers are selected. Example 2: Select 5 students from a population of 326, using a seed of 321. Enter 321 > STO > MATH > PRB (1) > ENTER. = 321rand ; Press ENTER. = 321. The random number generator is seeded with 321. Consider each student having and ID number from To select the sample of 5 students, enter MATH > PRB > 5 = randint ( enter 1>, > 326 >, > 5 > ENTER. = { scroll (>)> }. The five randomly selected students are students numbered 106, 243, 112, 203, 140. Note: Be sure to separate your entries of 1, N, n with commas. It is not necessary to close the parentheses of the randint entries..

7 TI-83/4 3 D. To select a systematic sample, it is first necessary to calculate a sampling interval i = N/n, rounded up. Then select only one random selection, as described in Part 2 above: MATH > PRB > 5 > enter only 1 >, > N (no n entry) > ENTER = first data element selected. Then add k to each selection until you have n selections. If a selection exceeds N, subtract N; the sampling frame is cyclic. Example 3: Select a sample of 5 students from a population of 326, using a systematic sampling method. Use a seed of 321. First calculate k = N/n; round up. i = 326/5 = 65.2 = 66; interval is 66. Seed 321: 321 > STO > MATH > PRB > ENTER. = 321rand > ENTER. = 321. The generator is seeded with 321. Enter MATH > PRB > 5 = randint(1 >, > 326 > ENTER. = 106. This is your random start, and the first student selected. Continue: (+) > 66 > ENTER = 172. (+) > 66 > ENTER = 238. (+) > 66 > ENTER = 304. (+) > 66 > ENTER = 370 > ( -) > 326 = 44. The 5 students selected systematically are students numbered106, 172, 238, 304, and 44. B. Tables.: LISTS are the devices used to store data in tabular form for further graphing and summarization. A draft of the table is required. Note: The TI-83/4 does not construct tables. Lists are not tables, just representations of data appropriate for further operations in the TI-83/4. II. Entering data into lists: There are two methods to enter data into lists. The first is to enter the data on the home screen and then storing them into a list. Enter 2 nd >{ (brace, above parenthesis), the enter each datum, separating each with a comma. After the last datum, enter STO > 2 nd > L1 = data is stored into List 1. You may use any list L1 L6. Most list entries in this manual use this approach. An alternative, perhaps quicker, method of listing data is to access the List Library directly to enter the data. Enter STAT > EDIT-1 > ENTER to access the library. Find an empty list and scroll to the 1 st position of the empty list and enter each datum followed by > ENTER. The cursor will automatically scroll down to the next space. Continue until you have entered all your data. The data is automatically stored in the list. To verify your lists at any time, enter STAT > 1(Edit) > ENTER to view your lists. All lists will appear.

8 TI-83/4 4 III. Clearing lists: To clear all lists, enter 2 nd > MEM > 4 (Clear all lists) > ENTER = DONE. All your lists are cleared. To clear an individual list, access the List Library, STAT > 1(Edit) > ENTER, then scroll to the heading of the list you wish to clear, enter CLEAR > ENTER. The individual list is cleared, while the other lists remain intact. 3. Sorting list data: To sort the data in any list, enter STAT > SORT A (ascending, i.e. lowest to highest) or SORT D (descending: highest to lowest) > (Enter the list number to be sorted) Lx. You may also use the 2 nd > LIST (above STAT) > OPS function for the same options. 4. Listing data in grouped data or frequency tables: If the x-data is constructed into classes, with lower and upper class limits, use the class midpoints or class marks, in descending order, for the first list. The second list is a listing of the frequencies related to each class. Enter the lists as above, identifying the lists as L1 to L6. You may use any list number, retaining previous lists for future use. If you store data on a list with previous data, the new data will override the old data. Keep track of your lists. C. Drawing graphs, based on lists. 1. Constructing a bar graph for ungrouped data: Remember, the first, usually nonnumeric, column is entered numerically in a list as 1, 2, 3, 4,... Press 2nd > STATPLOT > 1(Plot 1) >- ENTER = Plot 1 > ON > ENTER ; scroll down to TYPE > scroll to histogram icon (3) > ENTER; scroll down to Xlist : enter 2nd > List number of first column > scroll down to Freq : enter 2nd > List number of second column > After setting up the graph, press Z00M > 9(Zoomstat); the bar graph will appear. With the graph in view, if you press TRACE,, you will see the graph with related data for each bar. Scroll left or right (< >) to move from bar to bar. Example 4: From the following table, construct a bar graph. Oil reserves in the Western Hemisphere, by region, 1985, in billions of barrels. Region Reserves(BBbl) Central & South America 72.3 Mexico 64.3 United States 38.8 Canada 17.6

9 TI-83/4 5 To construct the bar graph for the above data in the TI-83/4, first make lists of the data in the 2 columns: Enter 2nd > { > 1 >, > 2 >, 3 >, > 4 > STO > 2nd > L 1 > ENTER ; Enter 2nd > { > 72.3 >, > 64.3 >, > 38.8 >, > 17.6 > STO > 2nd > L 2 > ENTER ; Enter 2nd > STATPLOT > 1 > On > ENTER > Type: (3rd icon) > ENTER > XList: > 2nd > L1 > Freq: 2nd > L2 > ZOOM > 9 2. Constructing a histogram for a frequency table. Begin as above: Press 2nd > STATPLOT > 1(Plot 1) >> ENTER = Plot 1 > ON > ENTER ; scroll down to TYPE > scroll to histogram icon (3) > ENTER; scroll down to Xlist : enter 2nd > List number of first column > scroll down to Freq : enter 2nd > List number of second column > Press WINDOW to set up the scale of the histogram: Xmin = 0 or lowest value in first class Xmax = highest value in last class or more Xscl = class width Ymin = - Ymax/4 ; note the negative. Ymax = highest value or more in second list or more Yscl = 1 (or more) Yres = 1 Note: you may store more than one plot. Use Plot 2 or other plots for new plots. However, make sure only the plot you are working on plot is ON. Shut other plots OFF. Otherwise the calculator will show all ON plots simultaneously. Example 5: From the following frequency table, construct a histogram. Frequency distribution of distances completed by 50 Los Angeles Marathon runners, 1990, to the nearest mile. x x m f

10 TI-83/4 6 First enter the x m (midpoint) column, not the x-column, as a LIST ; e.g., L3 Enter the f-column as a second LIST; e.g., L4. Be sure the f-value corresponds to the appropriate midpoint value. Then, enter 2nd > STATPLOT > #(choose a plot number) > ON; Scroll down to Type: select the histogram (3rd) icon > ENTER; scroll down to Xlist : enter 2nd > List number of x m > column (L3) > scroll down to Freq : enter 2nd > List number of f> column (L4) > Press WINDOW: Xmin = 0 Xmax = 30 Xscl = 5 Ymin = - 12/4 Ymax = 12 Yscl = 1 Yres = 1 Press GRAPH; the histogram appears. Press TRACE and scroll to each for class; values of class boundaries are shown as min and max, frequencies as n. D. Calculating summary measures, based on lists. 1. For ungrouped data tables: Press STAT > CALC > 1(1Var(able) Stats) > ENTER =1-VarStats; enter 2nd > List number of second column only > ENTER = mean, standard deviations, n, (scroll down) median, max, min, quartiles. 2. For frequency tables: Press STAT > CALC > 1(Variable) Stats) > ENTER =1-VarStats; enter 2nd > List number of 1st column (marks) >, > List number of 2nd column (freq.) > ENTER = mean, standard deviations, n, (scroll down) median, max, min, quartiles. Note: For frequency distributions, remember to enter 2 list numbers, separated by commas. Example 6: Find the mean, median and standard deviation for each of the tables above. 5. For the ungrouped, discrete, data table: Enter STAT > CALC 1 > ENTER = 1-VarStats( 2nd > List number of second column > ENTER = Mean (x-bar) = BBbl; Standard dev (s x ) = BBbls; scroll down Median = BBbl.

11 TI b. For the frequency table: Enter STAT > CALC >1> ENTER = 1-VarStats( 2nd > List number of x m column >, > List number of f-column > ENTER = Mean ( x ) = 12,7 mi. Standard dev (s x ) = mi.; scroll down Median = 13 mi. c. See Part II, Section A, for finding summary measures of probability or relative frequency tables. E. Boxplots: 1. For ungrouped data tables: Press 2nd > STATPLOT> #(Plot #) > ENTER = Plot #:> ON > ENTER > scroll down to TYPE > go to second row, use 2nd boxplot icon > ENTER > scroll down to Xlist; enter 2nd > List number of data > ENTER > scroll down to Freq; enter 1. Press ZOOM > 9; and/or TRACE 2. For frequency tables: Press 2nd > STATPLOT > #(Plot # > ENTER = Plot # > ON > ENTER > scroll down to TYPE > to second row, use first boxplot icon, with outliers > ENTER > scroll down to Xlist; enter 2nd > List number of 1 st (x) column > ENTER > scroll down to Freq; enter 2nd> List number of 2nd (f) column. Press ZOOM > 9, and/or TRACE. TRACE identifies five point values of boxplot.

12 TI-83/4 8 RELATIONS BETWEEN TWO (OR MORE) VARIABLES: LINEAR CORRELATION AND REGRESSION FUNCTIONS Linear correlation and regression determine if and how a relation exists between two (or more) variables in an experiment. For such problems, the TI-83/4 constructs the scatterplot for the data provided, calculates Pearson s r, the coefficient of determination r 2, provides the statistics necessary for the construction of the regression line of least squares (y = b 0 + b 1 x), constructs the regression line on the scatterplot, predicts the y-value for any x-value, and provides the standard error of estimate and other statistics necessary for establishing a prediction interval for any predicted y-value. It also calculates the critical t-value for a hypothesis test of correlation (H 0 : ρ = 0; i.e. there is no correlation; H 1 : ρ 0). A. Entering the data: For correlation and regression functions, all data related to the variables (x,y) are entered as lists, using the first list for the x-data, and the second list for the y-data. You may choose any combination of list labels (L 1 - L 6 ), but it is essential that the order of the elements in each list correspond to the order of the ordered pairs (x i, y i ). B. Constructing a scatterplot: After entering the x and y data in lists, e.g. L 1 and L 2, you can construct the scatterplot of the data using the STATPLOT function: Enter 2nd > STATPLOT > 1(Plot1) > ENTER > ON > ENTER: Type: highlight the 1st icon (scatterplot) Xlist:: L 1 - or whatever list you entered the ordered x-values. Ylist: L 2 - or whatever list you entered the ordered y-values. Mark: Choose any point marking method. Enter ZOOM > 9 (Stat): The scatterplot will appear with points shown. Enter TRACE and you will see the corresponding x and y values point by point. C. Determining and constructing the line of linear regression or LLS Presuming that you have previously entered the data in two lists, as in A above; Enter STAT > TESTS > E(LinRegTTest) > ENTER = LinRegTTest: enter Xlist: L 1 (or your list number for the x-values). Ylist; L 2 (or your list number for the y-values). Freq: 1 β and ρ: 0; this is related to the hypothesis testing operation. RegEQ: Enter VARS(variables) > YVARS(y>variables) > 1(Function) > ENTER > 1(Y 1 ) > ENTER = Y 1. These entries for RegEQ (regression equation) will store your regression line as Y 1. If you are using Y 1 for another purpose, choose another Y. Then, scroll down to CALCULATE > ENTER =

13 TI-83/4 9 LinRegTTest (results): y = a + bx. This is the TI-83/4 form of y = b 0 + b 1 x t: = The critical value for the hypothesis test. p = The P-value for the hypothesis test. df = n -2 (for one sample, two variables). a = b 0 value (intercept) b = b 1 value (slope) s = Standard error of estimate, used to determine interval of estimate. r 2 = Coefficient of determination value. r = Pearson s r value. You can now construct your regression line y = b 0 + b 1 x using the above data. D. Drawing the regression line: After you have constructed the data set and determined and stored your regression equation, the TI-83/4 will automatically draw the regression line onto the scatterplot. To see the line enter ZOOM > 9: the scatterplot will appear and the LLS will materialize. E. Predicting y from the regression equation. Presuming that all of the above has been done, notably the linear regression equation, you can now predict any value of y, given and value of x. The predicted coordinates areon the regression line. You may either manually calculate a predicted value of y, using the regression equation y = b 0 + b 1 x, with the values of b 0 and b 1 determined above, and insert your x-value. Or you may use the TI-83/4 to determine the predicted y-value from the stored regression equation. Enter VARS > YVARS > 1(function) > 1(Y 1 or whatever Y you used to store your equation) > ENTER =Y 1 > ( > enter x-value > ) > ENTER = predicted y-value, corresponding to entered x-value. Be sure to enter both parentheses in entering your x-value. F. Constructing a prediction or confidence interval for y: The prediction interval is an estimation procedure, using an interval to predict, at any given level of confidence, the true value of y using the predicted value of y as the best available estimator; i.e. at c, (y-hat E) < y < (y-hat + E), where n( x 0 x) 2 ( ) 2 E = (± t α/2 )(s e ) n + [ n x 2 x ]

14 TI-83/4 10 The calculator does not provide the prediction interval directly, but it provides the necessary data and statistics from the results of the STAT > TESTS > E operation, above, to calculate E and construct the prediction interval manually. Example 7: For a study of the relationship between the size and weight of U.S. brown bears, eight bears were captured, tagged, their weights (in pounds) and lengths from snout to tip of rear paws (in inches) recorded, and then released. Below is the data collected from each bear in the sample: Bear Length Weight Construct a scatterplot of the above data. What is the correlation, if any, between a bear s length and weight? What proportion of a bear s weight can be attributed to its length? Construct a regression equation for the above data, and predict the weight of a bear 70 inches in length. Also, what would be the true weight of a brown bear 70 inches in length? a. To enter data: Enter x-values, length, into L1. Enter y-values, weight, into L2. Be sure that the pairs of (x i, y i ) correspond to the same positions on the two lists. b. To construct the scatterplot: Enter 2nd > STATPLOT > #(use appropriate number) > ENTER > ON > Type: 1st drawing Xlist: L 1 Ylist: L 2 Mark: 1st icon, or any you choose. ZOOM > 9 = scatterplot of the above data. c. To determine correlation and determination coefficients and regression line: Enter STAT > TESTS > E > ENTER = Xlist: L 1 Ylist: L 2 Freq: 1 β and ρ: 0 RegEQ: VARS>YVARS>1 > ENTER > 1 > ENTER = Y 1 ; scroll down CALCULATE > ENTER =

15 TI-83/4 11 LinRegTTest t = p = df = 6 a = b = s = r 2 = r = Note: Do not round these values. You may round off final answers and conclusions after using these values. Conclusions: Correlation is , indicating a strong relation between a bear s length and weight. The coefficient of determination, r 2, indicates that approximately 80.5% of a bear s weight can be attributed to its length. Line of regression is y-hat = x, which has been stored as y 1. Note: You may store the equation in any unused y. To see line of regression, enter ZOOM > 9. To predict the weight of a bear 70 inches in length: Enter VARS > YVARS > 1 > ENTER = Y 1 > ( > 70 > ) > ENTER = lbs., or lbs. d. To construct a prediction interval: At a 95% level of confidence, construct a prediction interval for the true weight of a brown bear with a length of 70 inches. y = y-hat ± E; at x = 70 in., y-hat = lbs.(above) y = ± E n( x 0 x) 2 ( ) 2 Calculate E. E = (± t α/2 )(s e ) n + [ n x 2 x ] The data for the calculation of E is found in the LinRegTTest results above: n = 8; df = 6; s e = ; and From a t-table: at c = 0.95, α = 0.05; df = 6; t α/2,6 = ± 2.447; and The values of x, x and x 2 are found in the statistics of the x-values, List 1:

16 TI-83/4 12 Enter STAT > CALC >1 = I-Var Stats ( > 2nd > L 1 > ENTER: x = ; x = 516.5; x 2 = E = (± 2.447)( ) ( ) (516.5) 2 [ ] = ± = ± lbs. Note: Don t round until the final value or answer is determined. y = ± lbs. Conclusion: Based on a sample of 8 brown bears, we are 95% sure that the true weight of a bear 70 inches long could be anywhere between and lbs., or simply, between 150 and 500 lbs.

17 TI PROBABILITY FUNCTIONS A. Calculating parameters of probability functions Relative frequency or probability distributions represent the theoretical, probable distribution of population data. Population parameters summary measures for a probability or relative frequency distribution. If you have a probability distribution table with an x or x m column, an f- column and a P(x)- column entered into Lists, as described above, you can find parameters by using the STAT > CALC > 1 function as you did for calculating sample statistics, using the class midpoint and P(x) lists. Note that when a P(x) column is entered as a variable, the results show no s x value, but the σ x value is given. The x value is used for µ x. Example 8 (See Example 6): Find the population mean and standard deviation of the distances completed by all runners in the L.A. Marathon in Probability distribution of distances completed by 50 Los Angeles Marathon runners, 1990, to the nearest mile. x x m f P(x) First enter the x m - values into a list, e..g., L1, and P(x)- values into another list. e.g., L3, then enter STAT > CALC > 1> ENTER > 2 nd > L1 >, 2 nd > L3 > ENTER = µ x = x = 12.7 miles; σ x = 7.31 miles. All other summary values are population parameters also. Compare with sample statistics. B. Factorials (n!) Enter n-value; then press MATH > PRB > 4(!) > ENTER. Warning: The factorial function will not accept an n of 3 or more digits. It will show an OVERFLOW error; i.e. the resultant number is too large.

18 TI-83/4 14 C. Permutations: npr Enter n-value; then press MATH > PRB > 2(nPr) > r-value > ENTER. D. Combinations: ncr Enter n-value; then press MATH > PRB > 3(nCr) > r-value > ENTER. Note: Remember to enter the n-value first before going to the MATH-PRB function. E. Binomial distribution: 1. For single, discrete value of x a. Hard way: Using the Bernoulli binomial formula: : P(x) = ( n C x )(p x )(q n-x ) where p = P(S); q = P(F) = 1 - p. Enter n-value; then press MATH > PBR >3 > x-value > X(times) > p-value > ^(power) > x-value > X(times) > q-value > ^ (power) (n-x) value. Note use of multiplication (X) and exponent (^) symbols. Example 9: What is the probability of exactly 4 successes in 10 trials if the probability of success is 55%? a. Enter 10 > MATH >PRB > 3 > 4 > X(times) > 0.55 > ^ > 4 > X > 0.45 > ^ > 6 > ENTER = Note: Values of p and q should be entered as decimals; e.g. 55% = 0.55 or.55. The whole number 0 in your decimal value is not necessary, but the decimal point (.) is necessary. b. Easier way: Using the TI-83/4 binomial distribution function: 2nd > DISTR (above VARS). Enter 2nd > DISTR > 0 (binompdf) > n-value > p-value > x- value > ENTER. Note: Within the DIST function, the 0 option (scroll down to or enter 0) is for the binompdf function, the binomial probability density function of a single, discrete value of x or P(x). Be sure to use commas to separate n, p, and x-values.

19 TI-83/4 15 Example 10: Same problem as above, using binompdf function: Enter 2nd > DISTR > 0 = binompdf (10>, > 0.55., > 4 > ENTER. = For cumulative values of x; i.e. a range of values of x: Use the binomial cumulative density function: 2nd > DISTR > A(binomcdf) > n >, p >, > x > ENTER. Note use of commas. Example11: What is the probability of at most 4 successes in 10 trials if the probability of success is 55%? At most 4 means 4 (from 0 to 4 ). Enter 2nd > DISTR > ALPHA > A (or scroll down to A ) = binomcdf(: enter 10>, > 0.55 >, > 4 > ENTER. = = P(x 4). Example 12: What is the probability of at least 4 successes in 10 trials if the probability of success is 55%? At least 4 means 4 or more (up to the limit of trials, 10 in this exercise). P (at least 4) = 1 - P(x < 4) = 1 - P(0 or 1 or 2 or 3 ) = Enter 1 > (minus) > 2nd > DISTR > ALPHA > A (or scroll down to A ) = binomcdf (: enter 10 >, > 0.55>, > 3 > ENTER. = = P(x 4). Note: binomcdf means binomial cumulative density function of x, or P( all values less than or equal to x successes); in a binomial distribution this includes 0 successes). Example 13: What is the probability of between 4 to 8 successes in 10 trials if the probability of success is 55%? P( Between 4 and 8 (inclusive) ) means P[( x 8) - (x 3)]. Enter 2nd > DISTR > ALPHA > A (or scroll down to A ) = binomcdf(: enter 10 >, > 0.55 >, 8 > (-) (minus) > 2nd > DISTR > A = Binomcdf(; enter 10 >, > 0.55 >, > 3 > ENTER. = = P( 4 x 8). Note: We are using 2 binomcdf operations here. Remember to enter the close parentheses ) after the x-value of the first binomcdf. It is not necessary to close parentheses for one or the final operation of the binomcdf function. These binomial functions could also be performed using the binomial probability density function table found in the text, Table A-1. Remember, this table is a probability density function table; i.e. the probabilities of single, discrete values of x are given for selected values of n and p. For cumulative probabilities, individual probabilities must be added together. Other limitations of this table are a limited n (2-15) and limited p-values (in multiples of 0.10 only)

20 TI-83/4 16 NORMAL DISTRIBUTION FUNCTIONS Most inferences are made through the use of the standard normal distribution -- the z - distribution. The probability density function for a z-distribution can be found in the standard z-table, shown in your text as table A-2, also on the inside back cover, or in operations of the TI-83/4 2nd > DISTR > 2(normcdf) and 3(invNorm) functions. A. To find the probability of individual values of x or z in a normal distribution, use the 2nd > DISTR > 1(normpdf) function. 1. Example 14: Find P( z = 1.76) Enter 2nd > DISTR > 1 = normalpdf( : enter 1.76 > ENTER = Note: This was the probability of a z-value. 2. Example 15: Find P(x = 65 µ x = 70; σ x = 10). For a normal x-distribution, values of x, µ x and σ x must be entered. Enter 2nd > DISTR > 1 = normalpdf(: enter 65 >, > 70 >, > 10 > ENTER = B. To find the probability between any two values of z: i.e. Z 1, Z 2, use the 2nd > DISTR > 2(normcdf) function, entering the first and second z-values, from left to right. 1. Example 16: If Z 1 = 0; Z 2 = 1.76; find P(0 < z < 1.76): Enter 2nd > DISTR > 2 (normcdf) > ENTER = Normcdf(: enter 0 >, > 1.76 > ENTER = Notes: Two z-values must be entered, the minimum and maximum boundaries of the interval, reading from left to right, separated by a comma. The closing parenthesis ) need not be entered. 2. Example 17: Find P( < z < 0). Enter 2nd > DISTR > 2 (normcdf) > ENTER = normcdf(enter (-) > 1.76 >, > 0 > ENTER = Notes: Use the negative (-) key. Even though z-values may be negative, P(z) is always positive. 3. Example 18: Find P( < z < 1.76). Enter 2nd > DISTR >>2 (normcdf) > ENTER = normcdf( enter (-) > 1.76 >, > 1.76 > ENTER = = 2(0.4608)

21 TI-83/ Example 19: Find P( z < 1.76). This is the entire left half of the normal curve plus a portion from 0 to 1.76 on the right half; i.e. the left boundary of this area is negative infinity, the right boundary is Enter 2nd > DISTR > 2 (normcdf). ENTER = normcdf(: enter 2nd > EE > 99 >, > 1.76 > ENTER = Note: 2nd > EE > 99 = E99 = 10 99, an extremely large number used to stand for infinity. Correspondingly, -E99 stands for negative infinity. To enter -E99, enter: 2 nd > (-) > 2 nd > EE > 99. Remember to use a comma to separate z-values, including infinities.. 5. Example 20: Find P( z > 1.76). This is the right tail only, from z-value 1.76 to infinity. Enter 2nd > DISTR > 2 > ENTER = normcdf(: enter 1.76 >, > 2nd > EE > 99 > ENTER = C. To find the probability between any two values of x in a normal distribution, use the same functions as above, but remember to also enter values of µ x and σ x. Example 21: Find P( 60 < x < 80 µ x = 70; σ x = 10). Enter 2nd > DISTR > 2 = normcdf(; 60 >, > 80 >, > 70 >, > 10 > ENTER = Note: Remember to use commas. D. Finding z-values if the P(z) is known. This is the reverse of finding P(z). In using the z-table you would literally reverse the procedure above. With the TI-83/4, you would use the 2nd > DISTR > 3 (invnorm) -- inverse normal -- function. This function only reads probability values entered as a cumulative probability P( z Z x ), for any given ± Z x and/or P( x X x ), for any given ± X x, given µ x and σ x. 1. Example 22: What is the value of Z x for P( z Z x ) = This implies an area of the normal density function that is only on the left tail, the right boundary of which would be Z x Enter 2nd > DISTR > 3 (invnorm) = invnorm (: > ENTER = Example 23: What is the value of Z x for P( z Z x ) = This implies an area of the normal density function that is the entire left half plus a large portion of the right half, the right boundary of which would be Z x. Enter 2nd > DISTR > 3 = invnorm(: enter > ENTER = 1.36

22 TI-83/ Example 24: What is the value of Z x for P( z > Z x ) = This implies an area of the normal density function that is only on the right tail, the left boundary of which would be Z x. Remember, the invnorm function only reads probability values entered as a cumulative probability P( z Z x ), for any given ± Z x. Therefore, = = the density of the area Z x Enter 2nd > DISTR > 3 = invnorm (enter 1 > -(minus) > > ENTER = Or 2nd > DISTR > 3 = invnorm(: > ENTER = 2.25 E. Finding x-values if P(Z x ) is known. Use the invnorm function as above, but include the values of µ x and σ x. Remember, the invnorm function only reads probability valuesentered as a cumulative probability P( z Z x ), for any given ± Z x. Example 25: What is the test score which is the 90 percentile value for a distribution of scores with a mean of 70 and a standard deviation of 10? The 90 percentile implies a cumulative probability P(z Z x ) = 0.9. Enter 2nd > DISTR > 3 = invnorm(: 0.9 >, > 70 >, > 10 > ENTER = or 83 F. Assessing normality of data using a normal probability plot. Many data analysis methods presume a variable is normally distributed, or approximately so. The larger the sample, the more the data will present a relatively bell-shaped histogram or probability curve. On the other hand, for small samples, the shape of any frequency or probability graph is often uncertain, and often far from bell-shaped. A more sensitive graphic technique to assess normality is the normal probability plot, which plots each datum (x i ) and its corresponding z-score (z xi ) as a point on a set of x and z axes. The more linear the plot of (x, z x ) points appears, the more probable the set of data is normally distributed. To construct a normal probability plot, use the Statplot function: Enter the data into a List; then 2 nd > STATPLOT > go the any available Plot > ON > ENTER > (Be sure all other Plots are OFF) Scroll down to Type > scroll right to sixth (last) icon > ENTER > Scroll down to Data List: enter List number of data: 2 nd > L# > ENTER > Scroll down to Data Axis:: enter X > ENTER > Scroll down to Mark: select and enter the point style of your choice > ENTER > ZOOM > 9. Visually assess the linearity of the plot and hence the normality of the data. Note that the vertical z-axis has both positive and negative values

23 TI-83/4 19 THE CENTRAL LIMIT THEOREM AND ESTIMATION FUNCTIONS A. The Central Limit Theorem (CLT) allows the use of the normal distribution for any distribution of x, normal or non-normal. If the x-distribution is originally normal, the x -distribution is also normal. If the x-distribution is non-normal, the x -distribution can be assumed to normal for samples 30. In either case, µ x = µ x, and σ x = σ x n. If N is known and n/n 0.05, then σ x = σ x N n. The second factor, n N 1 is called the finite population adjustment or correction (FPA). N n N 1 ) In either case again, use the normal distribution functions, normcdf and invnorm, with σ x rather than σ x. B. Estimation is the process of determining an interval estimate for population parameters, µ x or p x. These estimates presume the use of a normal distribution by reason of the Central Limit Theorem. The primary functions for estimation are found in the STAT > TESTS functions of the TI-83/4. 1. Estimating µ x for large samples (n 30). Enter STAT > TESTS > 7(Zinterval) > Stats > ENTER = Zinterval: Scroll down to σ: Enter value of σ x ; x : Enter value of sample mean; n: Enter value of sample size; C-level: Enter value as decimal. Scroll down to Calculate > ENTER. = Zinterval ( x - E, x + E) or ( x E) < µ x < ( x + E) To find the margin of error E: E = ( x + E) - x Example 25: From a sample of 106 healthy persons, the average body temperature was 98.2 F and the standard deviation was 0.62 F. Find a 95% confidence interval for the body temperature of all healthy persons. What is the margin of error? Enter STAT > TESTS > 7 > Stats > ENTER = Zinterval: scroll down to: σ: 0.62 x : 98.2 n: 106 C-level: 0.95

24 TI-83/4 20 Scroll down to Calculate > ENTER. = Zinterval ( , ) i.e., F < µ x < F. Conclusion: Based on a sample of 106 persons, we are 95% certain that the average body temperature of healthy people in general is between and F. E = = F. 2. Estimating µ x for small samples (n < 30). There are two distinct situations here. If the original distribution of x is normal, use the z-distribution, regardless of the sample size. Use the same procedures as above. However, if the original x- distribution is not normal or cannot be presumed to be normal, the z-distribution cannot be used. The t- distribution is used instead. Estimating µ x for non-normal distributions w. small samples, using t-distribution: Enter STAT > TESTS > 8(Tinterval) > Stats > ENTER = Tinterval: Scroll down to: x : Enter value of sample mean; s x : Enter value of sample standard deviation; n: Enter value of sample size; C-level: Enter value as decimal. Scroll down to Calculate > ENTER = Tinterval ( x - E, x + E) i.e.: x - E < µ x < x + E To find E : E = ( x + E) - x Example 26: Same as Example 25 above, with n = 10. Enter STAT > TESTS > 8> ENTER > Stats > ENTER = Zinterval: scroll down to: x : 98.2 s x : 0.62 n: 10 C-level: 0.95 Scroll down to Calculate; ENTER. = Tinterval ( , ) i.e., F < µ x < F. Conclusion : Based on a sample of 10 persons, we are 95% certain that the average body temperature of healthy people in general is between and F. E = = F. 3. Estimating p x for large samples (n 30). The estimation of p x is based on the normal approximation of a binomial distribution, and can only be made with large samples. The observed probability of success, or sample probability, is p, called p-hat.

25 TI-83/4 21 Enter STAT > TESTS > ALPHA > A (or scroll down to A:1 variable > PropZInt) > ENTER > Stats > ENTER = 1-PropZInt: Scroll down to: x: Enter value of number of successes. Note: This is x, not x. n: Enter value of sample size; C-level: Enter as decimal value; Scroll down to Calculate > ENTER = pinterval( p - E, p + E). To find E: E = ( p + E) - p or ( p - E )< p x < ( p + E) Example 27: In a survey of 1068 U.S. households, 673 stated that they use a telephone answering machine or service. Find the 90% confidence interval of the total proportion of U.S. households that use telephone answering machines or services. p = 673/1068 = Enter STAT > TESTS > A > ENTER > Stats > ENTER = 1-PropZInt: Scroll down to: x = 673; n = 1068; C-level = 0.9; Scroll down to Calculate > ENTER = pinterval ( , ) i.e.: < p x < Conclusion: Based on a sample of 1068 households in the U.S., we are 90% confident that between 61% and 65% of U.S. households use telephone answering machines or services. E = = To use the finite population adjustment or correction in estimtion, enter (σ x ) > (X)(times) > 2nd > > ( > N > ( ) > n > ) > /(divide) > ( > (N 1) >). Note the use of parentheses. In estimating p x, the value of σ = p p q n must be entered manually. Example 28: If N = 500, n = 50, p = 0.68; estimate p x at c = Since 50/500 = 0.10, FPA must be used. Use hand calculation method. Calculate margin of error. Note the strategic use of parentheses here: 1.96 > (X)(times) > 2nd > ( > ( > 0.68 > (X) > 0.32 >) > /(divide) > 50 >) > X(times) > 2nd > > ( > (500 50) >) > /(divide) > 499 > ENTER = Therefore p x = 0.68 ± = (0.557, 0.803)

26 TI-83/4 22 Notes on estimation functions : 1. For estimation of µ x functions on the TI-83/4, the value of σ x, not σ x, is entered for the statistic σ. The TI-83/4 calculates σ x. If the FPA is required, enter σ x X(times) FPA. 2. For estimation of p x, samples must be 30. If FPA is necessary, hand calculation of the p q σ = is required; see Example 24. p n 3. The STAT - TESTS options can be entered directly, rather than scrolled down. For STAT -TESTS option A, you can enter ALPHA - A (above MATH), or scroll down. 4. The option of using STAT - TESTS >... > Data, rather that Stats, is based on the entry of raw data, i.e., observed values of x, into lists, either as an array or a frequency distribution, prior to the calculation of sample statistics. The use of STAT - TESTS >... > Stats requires the sample mean and standard deviation to be known.

27 TI-83/4 23 HYPOTHESIS TESTING FUNCTIONS The TI-83/4 functions for testing hypotheses vary according to the parameter being tested, the sample size(s) and the number of samples involved in the testing. For this course we will limit discussion to one sample testing procedures. We will show the traditional 7 steps and the P-value procedures. To test hypotheses of µ x or p x, based on samples of n 30, use the STAT - TESTS > 1(Z-Test) function for µ x or STAT > TESTS > 5(1 sample -PropZTest) function for p x. For testing hypotheses for µ x with samples less than 30, use STAT >TESTS > 2(T-Test) function. There is no test for p x with small samples. Note: These functions will identify the test statistic and the P-value for a test. They do not identify the critical value(s) nor make your decision to reject or not reject H 0. You must still set up the problem and decide from the results provided. A. Testing a hypothesis of µ x with n 30: Use STAT > TEST > 1(Z-Test) function. Example 29. For a sample of 106 adults, their average body temperature was 98.2 F, with a standard deviation of 0.62 F. Test the claim that the mean body temperature of healthy adults is 98.6 F. Use a level of significance of 5%. Traditional method: 1. H 0 : µ x = 98.6 F H 1 : µ x 98.6 F 2-tail test 2. At α = 0.05, with a 2-tail test, for a sample n 30, the critical values are z α/2 = ± Use z-table. 3. Reject H 0 if z t or z t Otherwise, do not reject. 4. Calculate z t : Enter STAT > TESTS > 1 > Stats > ENTER = µ 0 = 98.6; scroll down σ x = 0.62 x-bar = 98.2 n = 106 H 1 is µ µ 0 > ENTER; scroll down Calculate > ENTER = Z-Test: µ x = 98.6 z t = p = x ; x = 98.2 n = z t = < -1.96; Reject H 0.

28 TI-83/ Conclusion: Based on a test of a random sample of 106 adults, at a level of significance of 5%, there is very strong evidence that the mean body temperature of healthy adults is not 98.6 F, but may be less, based on the data available. P-value method: P-value is p = x = which is extremely significant. Therefore, reject H 0 without doubt. Notes: 1. The calculator is only useful for Step 5; you must set up the problem manually beforehand. C. If you choose the P-value method, you must still use at least Steps 1 and 2. D. You may also use the DRAW option to show the test results, rather than the CALCULATE option. It will show a (norrmal) z or t-curve, with only the values of the test statistic and P-value given, and the area of the P-value highlighted. The CALCULATE option is sufficient for this ciurse. B. Testing a hypothesis of µ x with one sample of n < 30: Use STAT > TEST > 2 (T-Test) function. Example 30. A sample of seven 12-ounce aluminum cans was subjected to a pressure test. The amount of pressure, in lbs/sq.in., that each can withstood before crumbling was:270, 273, 258, 204, 254, 228, 282 lbs. At a significance level of 0.01, test the claim of the manufacturer that these cans can withstand a mean pressure of over 250 lbs/sq.in. Note: These are data, not statistics being provided. Traditional method: 1. H 0 : µ x 250 lbs./sq.in. H 1 : µ x > 250 lbs/sq.in. right tail test 2. At α = 0.01, with a right tail test, for a sample n < 30, the critical value, at d.f. = 6, is t α = Use t-table 3. Reject H 0 if t t Otherwise, do not reject. 4. Calculate t t : First enter the above data into List 1: Enter 2nd > { > 270 >, > 273 >, > 258 >, > 204 >, > 254 >, > 228 >, > 282 > STO > 2nd > L 1. Clear, and then enter STAT > TESTS > 2 > Data. ENTER = µ 0 = 250 List = L 1 Freq = 1 H 1 is µ > µ 0 > ENTER CALCULATE > ENTER =

29 TI-83/4 25 T-Test: µ x > 250 t t = p = x = s x = n = t t = < 3; Do not reject H Conclusion: Based on a test of a random sample of 7 cans, at a 1% level of significance, there is insufficient evidence to reject Ho. The mean sustaining pressure is less than or equal to 250 lbs./sq.in. The manufacturer s claim is not supported, based on this test. P-value method: p = ; extremely insignificant. We cannot reject H o. There is very little evidence to support the manufacturer s claim. C. Testing a hypothesis of p x with one sample of n 30: Use STAT > TEST > 5 (1-PropZTest)function. Testing of hypotheses with p x require large samples to support the normal approximation of a binomial distribution. Example 31: A random sample of 100 cola drinkers took a taste test. They were asked to choose the better taste between Coke and Pepsi 1; 51 preferred Coke. At a 5% level of significance, test the claim that Coke is preferred over Pepsi 1 by more than 50% of cola drinkers. Note: We re back to statistics being provided here. Traditional: 1. H 0 : p x 0.5 H 1 : p x > 0.5 right tail test 2. At α = 0.05, with a right tail test, the critical value, is z α = Use t-table 3. Reject H 0 if z t Otherwise, do not reject 4. Calculate z t : Enter STAT > TESTS > 5 > Stats > ENTER; p 0 = 0.5 x = 51; Note: This is x, not x-bar. n = 100 H 1 is prop > p 0 > ENTER CALCULATE > ENTER = 1-PropZTest: prop > 250 z t = 0.2 p =

30 TI-83/4 26 p = 0.51 n = z t = 0.2 < 1.645; Do not reject H Conclusion: Based on a test of a random sample of 100 cola drinkers, at a 5% level of significance, there is insufficient evidence to reject H 0. The proportion of drinkers who favor Coke is less than or equal to 50%. Coke s claim is not supported by these data. P-value method: p = ; extremely insignificant. We cannot reject H o. There is very little evidence to support Coke s claim.

31 Florida A & M University (FAMU) H. Biske Revised; Dec., 1999 USING THE TI-83/4 STATISTICAL FUNCTIONS The Texas Instruments TI-83/4 calculator is required equipment for students taking this course in Statistics. It is the best calculator for statistical functions, surpassing other models and even later TI models. Instruction will be provided in class on the use of the TI-83/4 to calculate certain statistical functions. Instruction will not be provided for other calculators or computer functions. * In the following descriptions and throughout this booklet a dash is used to show a sequence of key strokes or entries. For example: 2 nd > MEM > 3 > ENTER indicates four keystrokes. The subtraction operation will be shown with parentheses ( - ). Before using the TI-83/4, familiarize yourself with the keyboard: the basic arithmetic and scrolling operations in blue, the ALPHA function in green, and particularly the 2nd functions, indicated in yellow. The 2nd functions particularly are used heavily in statistical functions. Also, before using the calculator, clear any existing data that may have been stored from previous use. To clear the TI-83/4 memory enter 2nd > MEM > 3(Clear entries) > ENTER; and 2nd > MEM > 4(Clear lists) > ENTER; and 2nd > MEM > 5 (Reset) > 2 (Default). To go to the home screen for new operations, press the CLEAR button or 2nd > QUIT. I. DESCRIPTIVE STATISTICS FUNCTIONS A. Generating random numbers: The TI-83/4 s random numbers generator can help you select a random sample of size n from a population of size N. 1. Seeding the random numbers generator establishes the same starting point for a group of persons using separate TI-83/4s; it will ensure that each person gets the same random numbers. It is not required for use of the random number generator for individual use. To set a seed, enter any number other than 0 and store it in the random number generator. Example 1: Seeding with 123 : Enter 123 > STO > MATH > PRB (rand) > ENTER = (resulting in) 123 rand; press ENTER = 123. The random generator is now seeded at 123. Note: Seeding is required for each use of the random number generator, if seeding is desired.

32 Florida State University (FSU) H. Biske Revised; Dec., 1999 USING THE TI-83/4 STATISTICAL FUNCTIONS The Texas Instruments TI-83/4 calculator is required equipment for students taking this course in Statistics. It is the best calculator for statistical functions, surpassing other models and even later TI models. Instruction will be provided in class on the use of the TI-83/4 to calculate certain statistical functions. Instruction will not be provided for other calculators or computer functions. * In the following descriptions and throughout this booklet a dash is used to show a sequence of key strokes or entries. For example: 2 nd > MEM > 3 > ENTER indicates four keystrokes. The subtraction operation will be shown with parentheses ( - ). Before using the TI-83/4, familiarize yourself with the keyboard: the basic arithmetic and scrolling operations in blue, the ALPHA function in green, and particularly the 2nd functions, indicated in yellow. The 2nd functions particularly are used heavily in statistical functions. Also, before using the calculator, clear any existing data that may have been stored from previous use. To clear the TI-83/4 memory enter 2nd > MEM > 3(Clear entries) > ENTER; and 2nd > MEM > 4(Clear lists) > ENTER; and 2nd > MEM > 5 (Reset) > 2 (Default). To go to the home screen for new operations, press the CLEAR button or 2nd > QUIT. I. DESCRIPTIVE STATISTICS FUNCTIONS A. Generating random numbers: The TI-83/4 s random numbers generator can help you select a random sample of size n from a population of size N. 1. Seeding the random numbers generator establishes the same starting point for a group of persons using separate TI-83/4s; it will ensure that each person gets the same random numbers. It is not required for use of the random number generator for individual use. To set a seed, enter any number other than 0 and store it in the random number generator. Example 1: Seeding with 123 : Enter 123 > STO > MATH > PRB (rand) > ENTER = (resulting in) 123 rand; press ENTER = 123. The random generator is now seeded at 123. Note: Seeding is required for each use of the random number generator, if seeding is desired.