Statistical Calculations and Tests Using the TI 83/84.

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1 Statistical Calculations and Tests Using the TI 83/84. The document is meant to be read with a calculator in hand. Work an example to see the results of every step. The content is in the order that the topics are covered in many "Elementary Statistics" texts so you can move through this document as each new topic is covered in class. Conventions and Advice. For brevity, the TI 83+ and TI 84+ will be referred to as TI84. The slight differences between the two calculators are noted where they occur. Many calculator functions require the use of a nd key. Since the need to use this key is obvious from the keyboard color code, it is not described in these instructions. Conditions for a valid test and interpretation of the results of the test are not discussed in this document. This you will learn in the classroom and by reading the text. Some of the Tests provide for a choice between using Data (stored in a list) and Stats (where you input the statistics of each sample). Many of your homework questions give you the statistics saving you the labor of entering the data. Most calculator operations require pushing the ENTER key to execute. This command is omitted from the description. If in doubt, hit ENTER. Many of the tests have an optional Draw output that provides a graphical depiction of the results and can aid understanding. Only the Calculate option is described. Using Stat Plot: make sure that only one of the plots is turned on at one time and that no function equation (Y=) is active (indicated by a highlighted equals sign) unless you want it to be. If you get a mysterious error message when using Stat Plot, it is most likely because you haven't precisely asked for what you want. The Basics - Working with lists. Most uses of the TI84 for statistical calculations require that you store data in lists. The easiest way to enter values into a list is to see the list displayed in a vertical column using STAT/EDIT/ to access the Stat Editor. The initial setup of Stat Editor displays lists L1 through L6 only. You can add additional columns for displaying lists by moving the cursor up to the name of the last-named list (L6 to begin with) and then move the cursor right to the next column. To insert a new column, go to the header position (list name) of one of the named columns and press INS. This will add a column to the left of the one from which you made the request. To delete one of the lists displayed in Stat Editor, go to header position of that list and press DEL. 1

2 You begin the next list by entering a name followed by ENTER. Data is entered into a list starting at the top row under the name, with an ENTER after each data item. When you're done with Stat Editor, select QUIT so that you don't inadvertently make unintended entries into lists. Stat Editor is not the "keeper of lists". It only displays them and allows you to enter data. Deleting columns in the Stat Editor will not delete a list. The complete list of all lists currently defined may be found by using LIST/NAMES. Deleting lists must be done using MEM/ [Mem Mgmt/Delete], which allows you to selectively delete variables including lists. MEM/4 [ClrAllLists] doesn't delete list names, only their content. Lists can also be entered by typing a left curly bracket, the entries in the list separated by commas, and ending with a right curly bracket as follows: {entry 1, entry,..}sto> and then name the list. For example, {1,,3,4}STO> T1 will store those 4 values into a list named T1. If you wish to enter the name of a list, it must be selected from the LIST/NAMES menu unless it is one of the lists L1, L, L3, L4, L5, L6 which are selectable from the keyboard. For example, if you wish to display the contents of a list without using the Stat Editor, use LIST/NAMES/ and select the name of the list followed by ENTER. To restore the Stat Editor to its factory setting of Lists 1 6 as the only lists displayed and in order from left to right, use STAT/5 and ENTER. This does not delete lists; it only modifies (or resets) what lists are displayed in the Stat Editor. Chapter Summarizing and Graphing Data 1. Grouping Data. To group data, you scan a list and tally the number of entries in the list that are within a list of ranges (or classes). The tally produces frequencies of occurrence and the result is called a frequency distribution. The easiest way of producing a frequency distribution using the TI84 is to first produce a histogram (see below). If you wish to do it manually, it is easiest to sort the list first. Choose STAT/EDIT/ and name the list you wish to sort. If the classes are single valued rather than a range of values, enter the class values in one list and the corresponding frequencies in another. Starting with a list of frequencies for each class, we can use List math to obtain a relative frequency distribution. For example, if the frequency data is stored in L, LIST/MATH/5: sum(l) will display the sum of the frequencies and then L/ANS STO> L3 will divide each element of L by the sum of the frequencies and store the relative frequencies in L3. The key ANS will always insert in its place the value of the last calculation displayed.. Plotting Histograms. A histogram of ungrouped data can be obtained using STAT PLOT. Select one of the 3 Plots to turn On. Choose graph type #3 (the third icon), specify the xlist and Freq:1. Then ZOOM/9 GRAPH will set the window parameters and produce the histogram. The TI84 chooses the class ranges and you can use TRACE to see what they are. You most likely won't like them. To set the window parameters yourself, press WINDOW, enter the lower limit of the first class as Xmin,

3 a value larger than the upper limit of the last class as Xmax, and the class width as Xsc1. Ymax must be large enough to show the largest class frequencies. If you need to produce a frequency distribution, simply TRACE the histogram and read off the class limits and frequencies. If a frequency distribution is given, a histogram may be plotted by entering the list containing class midpoints as the Xlist: and the list of frequencies as the Freq: 3. Other Statistical Graphics. The TI84 can be used to produce most, but not all, of the other graph types. The frequency polygon is STAT PLOT/Graph Type. The Ogive graph is produced also using Graph Type but using cumulative frequencies. If the frequencies are in L, the instruction LIST/OPS/6: cumsum(l) STO> L3 will store cumulative frequencies in L3. Dotplots can be produced with some cleverness and stemplots require a special program. Scatterplots are often used in statistics and are the TI84 Graph Type 1. Chapter 3 Descriptive Measures. 1. One variable statistics. Use STAT/CALC/1: 1 Var Stats Listname produces the sample mean and standard deviation of the data in the list, as well as the 5 number summary minx, Q1, Median, Q3, maxx. It also produces standard deviation according to the population formula and identifies it as σ x. Alternatively, these statistics can be computed individually by using LIST/MATH/ and selecting the relevant measure. Be aware that there are many different methods of computing quartiles and the results differ slightly. The method used in the TI84 is the same as the one used in Bluman. Excel's method is different. If the sample is in the form of a frequency distribution, store the midpoints in one list, say L1, and the frequencies in another, say L. Then 1-Var Stats L1, L will produce the sample statistics including mean and standard deviation.. Boxplots (graphical display of the 5-number summary). Use STAT PLOT/, select the plot #, turn it On, select the 4 th (modified Boxplot) or 5 th graph type, name the list containing the data, and ZOOM/9. The modified Boxplot displays outliers separately. Chapter 4 Probability 1. Simulations. The TI84 can be used to simulate samples from uniformly distributed integer populations. Use MATH/PRB/5: randint(lowest integer, largest integer, number in the sample) STO> listname.. Counting. Factorials, combinations and permutations are all computable from MATH/PRB/select from the list. For example, to compute 10 C 4, enter 10/MATH/PRB/3/4 to get 10 as a result. Chapter 5 Discrete Probability Distributions. 3

4 1. One-variable statistics. Given a probability distribution P(x) for a discrete random variable x, we can calculate the parameters of the distribution. Assume that all possible values for x are stored in L1 and the probabilities for each value are in L. STAT/CALC/1: 1-Var Stats L1, L will display the population mean, standard deviation and 5 number summary.. Probabilities for a random variable having a binomial distribution. P( x) = DISTR/binompdf(n, p, x). P(x x o ) = DISTR/binomcdf(n, p, x o ). P(x > x o ) = 1 P(x x o ). To calculate the complete probability distribution, use DISTR/binompdf(n,p) (leave out the x). Store these values to a list for later use. 3. Simulating a Binomial Random Variable. MATH/PRB/7 > randbin(n, p, N) STO> L1 will generate N instances (or observations) of the number of successes in an n trial binomial experiment where p is the probability of success on each trial. Chapter 6 Normally Distributed Variables. 1. Simulating a Normally Distributed random variable. MATH/PRB/6: randnorm(μ, σ, N) STO> L1 will generate N observations of a normally distributed variable having mean μ and standard deviation σ and store them in L1.. Areas under a normal curve. DISTR/: normalcdf (lower bound, upper bound, [μ, σ]) produces the area under a normal curve having mean μ and standard deviation σ between the lower and upper bound. If you leave out [μ,σ], you get the area under the standard normal curve. 3. Finding an x value for a specified area to the left. DISTR/3: invnorm(area to left, [μ,σ]). Again, you can leave out [μ,σ] for the standard normal curve. If you want x for a specified area to the right, use invnorm((1 area to the right), [μ,σ]) 4. Normal probability plots. Use STAT PLOT, choose one to turn on, select the last icon of the 6 graph types, specify the name of the list containing the data, choose X as the axis to represent the data (meaning normal scores will appear on the Y axis) and your choice of mark. ZOOM/9 will satisfactorily set the window parameters. Chapter 7 Estimates and Sample Sizes 1. Estimating a Population Proportion ( p). To use the normal distribution approximation to a binomial distribution, check that the number of successes x and the number of failures n-x in the sample are both 5 or greater. Obtain z α/ from DISTR/3:invNorm(1 α/) and compute the confidence interval from pˆ E to pˆ E where pˆ is the sample proportion x/n, E z / pˆ,and pˆ is estimated from pˆ 1 pˆ n. p ˆ / Alternatively, use STAT/TESTS/1 PropZInt. Enter x, the number in the sample having the specified attribute, n, and the confidence level (1 α). The output is the confidence interval for the population proportion p. 4

5 Determining the sample size. Since the margin of error is E z p1 p/ n ˆ / minimum sample size for a given margin of error is obtained by solving for n: z / n pˆ 1 pˆ where pˆ is the sample proportion used as a point estimate of E the population proportion p. If you are unable to make a reasonable guess of pˆ in advance of obtaining the sample, then use the maximum value of pˆ (1 pˆ ) which is ¼. Since n must be an integer, round the computed value up to the next integer.. Estimating a population mean (σ known). Compute the confidence interval from [ x E ] where E z / / n. z / can be obtained from DISTR/3:invNorm(1 α/). ˆ, the Alternatively, use STAT/TESTS/Z Interval. Select between Data or Stats. Specify σ, name the list containing the sample data, set Freq: 1, and specify the confidence level (1 α). The output displays the confidence interval as a lower and upper bound, and the mean, standard deviation of and number of data points in the sample. 3. Estimating a population mean (σ unknown) Compute the confidence interval from [ x E ] where E t / s / n. t / can be obtained, on the TI 84+ only, from DISTR/4:invT(1 α/, df) where df = n 1. Since the TI 83+ does not have an inverse T distribution, it's easiest to use Table F to find t /. Alternatively, use STAT/TESTS/T Interval. The data entry and selections are the same as the one sample z Interval except that we don't input σ and the t distribution is used instead of the normal distribution. 4. Sample size. If σ is known, the minimum sample size for a given margin of error is z / the next whole number greater than n. You can obtain z α/ from E DISTR/3: invnorm(1 α/). If σ is unknown, you would need an estimate of it. ( n 1) s 5. Estimating a population variance. The variable has the chi-square distribution with degrees of freedom (df) = n-1. For a confidence level 1 α, the variance confidence interval can be computed from ( n 1) s ( n 1) s, where χ L R L and χ R are the values of χ for which the chi-square distribution has area α/ to the left and to the right respectively. These values can be found from Table G or using the calculator's SOLVER to solve the equation cdf (0, X, df ) A where A = α/ for X = χ L and A = 1-α/ for X = χ R. The confidence interval limits for the standard deviation are just the square roots of those for variance. 5

6 Chapter 8 Hypothesis Testing This section describes only the P-value method. The methods based on critical values or confidence intervals use calculator methods described previously. 1. Testing a claim about a population proportion. H 0 : p = p 0, H 1 : p p 0, p > p 0, p < p 0. The probability of obtaining a simple random sample of size n which includes x members of the population having the attribute of interest would be given by the binomial distribution if the number in the population is much greater than the number in the sample. Assuming the null hypothesis is true, the P-value for a lefttailed test would be binomcdf(n,p 0,x) and 1 binomcdf(n,p 0,x) for a right-tailed test. For a two-tailed test, the P-value is just double the area in the smaller of the two tails. In every case, the null hypothesis is rejected if P α; that is, the sample is sufficiently unlikely assuming the null hypothesis is true. To use the normal distribution as an approximation to the binomial, first check that the number of successes np 0 and the number of failures n(1 p 0 ) are 5 or greater. Use STAT/TESTS/5:1 PropZTest. Enter p 0, x and n, and the form of H 1 (, <, >). The pˆ p0 output gives a restatement of H 1, the value of the test statistic z0, p0 1 p0 / n and the P-Value. It also reports the value of pˆ and n. If P < α, reject the null hypothesis p = p 0. Otherwise, don't.. Testing a claim about a mean (σ known). Use STAT/TESTS/Z Test. Select between Data or Stats. Specify μ 0, σ, name the list containing the sample data, set Freq:1, and choose the form of H 1. The output displays the form of H 1, the test statistic z 0 for this sample, the P-value, the mean and standard deviation of, and the number of data points in, the sample. For a confidence level α, reject H 0 if P < α; otherwise don't. 3. Testing a claim about a mean (σ unknown). Use STAT/TESTS/T Test. The procedure is the same as the Z test except that we don't input σ and the t distribution is used instead of the normal distribution. 4. Testing a claim about a standard deviation or variance. Compute the test statistic χ assuming the null hypothesis is true. The P-value is obtained from DISTR/ χ cdf (lower bound, upper bound, df) to obtain tail area. The tail area is area to the right of the test statistic for a right-tailed test, area to the left for a left-tailed test, and twice the area in the tail bounded by the test statistic for a two-tailed test. Chapter 9 Inferences from two samples 1. Inferences about two proportions. We use the normal approximation to the binomial which requires that the samples are independent and each sample has at least 5 with the specific attribute and at least 5 without. Use STAT/TESTS/6: -PropZTest for testing the hypothesis that the two population proportions are equal, or -PropZInt for a confidence interval for the difference between the two population proportions. 6

7 . Inferences about two means: independent samples. H 0 : μ 1 = μ ; H 1 : μ 1 μ, μ 1 > μ, or μ 1 < μ. Use STAT/TESTS/ SampTTest and choose between Data and Stats. For the Data choice, specify the name of each list, choose 1 for both Freq1 and Freq, choose the form of H 1, and choose between pooled and non pooled. Use the "pooled" t procedure if you believe the two populations have nearly equal standard deviations and "non pooled" otherwise. The output includes a restatement of H 1, the value of t assuming H o to be true, the P-value, the degrees of freedom s 1 / n1 s / n s / n s / n 1 0, s / n s / n 1 x x, the means and standard deviations of, and the number of 1 1 n1 1 n 1 data points in, each sample. If P α, reject the null hypothesis. Otherwise, don't. STAT/TESTS/ SampTInt will calculate a confidence interval for μ 1 μ, given a confidence level 1 α. 3. Inferences from matched pairs. Form a list of differences d = individual differences between the two values in a single matched pair e.g. L1 L STO> L3 if the pairs of data points are stored in L1 and L. Then use STAT/TESTS/ T-Test for a test of hypothesis μ d = 0 or TInterval to obtain a 1-α confidence interval for μ 1 μ. 4. Comparing variation in two samples. Use STAT/TESTS/-SampFTest. If needed, you can most easily find the critical value of F using Table H since the inverse F is not programmed. 1 Chapter 10 Correlation and Regression 1. Scatter plot STAT PLOT/1 > select ON, first graph type, specify the names of the xlist and ylist and select a mark. Then ZOOM/9 and GRAPH. Visual examination of the scatterplot should confirm that the points approximate a straight line pattern.. Calculating the components of the regression coefficient (if it should prove necessary). Use STAT/CALC/ Var Stats, xlist, ylist. The results enable you to calculate these components of the linear regression coefficient. x x x nx S yy y y y ny and S xx, S xy x x y y xy nxy. Note that S xy r. S S 3. Complete linear regression. Use STAT/CALC/ LinRegTTest and enter xlist, ylist, and the name Y 1. The name Y 1 is obtained from VARS, select Y VARS, Function, Y 1. As an output, TI84 will deposit the regression equation a + bx next to Y 1 = so it is ready to use or graph. The value of the test statistic t and the associated P-value are given. If the P-value < α, the null hypothesis (no linear correlation) is rejected, and xx yy 7

8 the alternative of a linear correlation is concluded. Other outputs of LinRegTTest are a, b, r and r. Alternatively, test r against the critical values of r listed in Table I. You can see the regression equation plotted against the scatter plot background (described in #1) by simply pressing GRAPH. 4. Variation and Prediction Intervals The standard error of estimate s e ( y yˆ) is an output of the LinRegTTest or can be calculated from s S (1 r ) /( n ). The margin of error for a value of the independent variable x = x o is e yy E t x) / S. / se 11/ n ( x0 The test LinRegTInt (TI 84+ only) calculates a confidence interval for the slope β of the regression equation. If the interval contains zero, there would be insufficient evidence to conclude that a linear correlation exists. xx Chapter 11 Multinomial Experiments and Contingency Tables 1. χ distribution The area under the χ -curve between the values lower bound and upper bound is given by DISTR/ χ cdf (lower bound, upper bound, df). It would be nice if the TI84 included an INV χ function but it doesn't. So to find the value of χ α such that the area under the curve to the right of that value is α, you can use the SOLVER as described in Chapter 7 #5. Alternatively, use the tabulated values of χ α in Table G.. χ Goodness of Fit Test. Store observed frequencies and expected frequencies in two lists, say L 1 and L. df is 1 less than the number of different values the random variable can have. Calculate the value of χ o = sum((l1 L) /L) and find the P Value = P( o ) from DISTR/ cdf ( o, E99, df ). The null hypothesis (the observed values have the distribution given by the expected frequencies) is rejected if P Value < α. Alternatively (TI 84+ only), use STAT/TESTS/χ GOF-Test. 3. way (contingency) tables. Begin with a way table stored in a matrix (say [A] ). Then select STAT/TESTS/ χ Test. Name the matrix containing the table of observed frequencies ([A]) and the matrix that will contain the table of expected frequencies. Outputs are χ,p, and df. The expected frequencies are calculated assuming non association (independence) of the row and column variables. If P < α, the null hypothesis (independence) should be rejected. Chapter 1 One-Way Analysis of Variance (ANOVA). ANOVA is particularly easy with this calculator. Simply enter the data in separate lists for each sample and choose ANOVA from STAT/TESTS. You must name the lists as arguments as in ANOVA(L1,L,L3). 8

9 Chapter 13 Nonparametric Statistics. 1. Sign Test. We are concerned with the proportion p of the less frequent sign. The null hypothesis is that p = 0.5. We can calculate the probability of obtaining a sample with x instances of the less frequent sign using the binomial distribution as in P = binomcdf(n, 0.5, x). Double this value for a two-tailed test. Unlike EXCEL, the TI84 lacks a command to RANK a data value in a list. For this reason, it is much preferable to use EXCEL for all of the following nonparametric methods involving ranks. Here are some ideas for using TI84.. Wilcoxon Signed-Ranks Test for Matched Pairs. Store the pairs of data in lists L1 and L. Store the differences in L3: L1 L STO> L3 and the absolute value of the differences in L4: abs(l1-l) STO> L4. Now sort the lists according to the ascending order of L4: SortA(L4,L3). You now have the differences sorted in order of their absolute values. It is easy now to collect in one list (say L5) all of the positive ranks and in another list (say L6) all of the negative ranks. Sum both lists (LIST/MATH/5) and let w s be the smaller sum. Then complete the rest of the procedure on p Wilcoxon Rank-Sum Test. The TI84 can be used to find the ranks. If each of the two samples are stored in lists, say L1 and L3, we can concatenate the lists using LIST/OPS/augment(L1,L3) STO> L5. Now sort L5 in ascending order using LIST/OPS/1. Create two new lists, say L and L4, by listing, for each sample value, its corresponding rank obtained from the sorted L5. Remember the way of treating rank ties: assign the mean of the ranks involved in the tie to each data value so ranked. Then sum the ranks of either sample, give it the name R, and continue with the procedure on p Kruskal-Wallis Test. The initial work is the same as for the Wilcoxon Rank-Sum Test but with three or more lists. Form the "giant list" in the same way using augment (L1,L, L3,.) and sort this list. Then you must tediously form the lists of ranks for each sample and sum each of these rank lists to form R 1, R, etc. 5. Rank Correlation. Determine the ranks of each value in each of the lists forming the sample pairs. Sample pairs may not be disassociated with each other. Form two additional lists, replacing the values in the sample pairs with their ranks. The TI84 is not very useful here except perhaps in sorting to establish rank. However, the rank correlation coefficient r s may be found by finding the linear correlation coefficient r for the paired lists of ranks, say using LinRegTTest. 9