Calculate Confidence Intervals Using the TI Graphing Calculator, Lecture notes of Statistics

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Calculate Confidence Intervals Using the TI Graphing Calculator Confidence Interval for Population Proportion p Select: STAT / TESTS / 1-PropZInt x: number of successes found in sample n : sample size C-Level: 0.90, 0.95, 0.99, etc Calculate: Select Calculate and press Enter Program Output: Confidence Interval: (lower bound, upper bound) Calculated value of p-hat statistic Sample size n Confidence Interval for Population μ (σ is known) Select: STAT / TESTS / ZInterval Inpt: Use arrow keys – select Stats σ : value of population standard deviation x: value of sample mean statistic n : sample size C-Level: 0.90, 0.95, 0.99, etc Calculate: Select Calculate and press Enter Program Output: Confidence Interval: (lower bound, upper bound) Value of x-bar statistic Sample size n Confidence Interval Population μ (σ is unknown) Select: STAT / TESTS / TInterval Inpt: Use arrow keys – select Stats x : value of the sample mean statistic sx: value of sample standard deviation statistic n : sample size C-Level: 0.90, 0.95, 0.99, etc Calculate: Select Calculate and press Enter Output: Confidence Interval: (lower bound, upper bound) Value of the sample x-bar statistic Value of the standard deviation statistic s Sample size n Confidence Interval Population σ The current versions of the TI graphing calculators do not have a program to calculate confidence intervals for σ2 and σ. There is a program named S2INT that can be installed on a TI. See page 383 of your text book for details. CI for Difference of Population Proportions ( p 1 – p 2 ) Select: STAT / TESTS / 2-PropZInt x1: number of successes found in first sample n1 : sample size of first sample x2: number of successes found in second sample n2 : sample size of second sample C-Level: 0.90, 0.95, 0.99, etc Calculate: Select Calculate and press Enter Program Output: Confidence Interval: (lower bound, upper bound) Calculated value of p-hat statistic of first sample Calculated value of p-hat statistic of second sample Sample size n of first sample Sample size n of second sample CI for Difference of Population Means ( μ 1 – μ 2 ) Population σ's are unknown, but we assume σ 1 = σ 2 . Sample statistics are taken from two independent simple random samples. Select: STAT / TESTS/ 2-SampTInt Inpt: Use arrow keys – select Stats x1: sample mean statistic of first sample sx1 : sample standard deviation of first sample n1 : sample size of first sample x2: sample mean statistic of second sample sx2 : sample standard deviation of second sample n2 : sample size of second sample C-Level: 0.90, 0.95, 0.99, etc Pooled: Select Yes (We are assuming σ 1 = σ 2 ) Calculate: Select Calculate and press Enter Program Output: Confidence Interval: (lower bound, upper bound) Degrees of freedom used for t-distribution Mean of first sample Mean of second sample Standard deviation of first sample Standard deviation of second sample Pooled sample standard deviation Sample size of first sample Sample size of second sample 1 Hypothesis Tests Using the TI Graphing Calculator (pages 2 - 4) Hypothesis Test for Population Proportion p Select: STAT / TESTS / 1-PropZTest po : the population proportion stated in H o x: number of successes found in sample n : sample size prop ≠ po < po > po (select H 1 test type ) Calculate: Select Calculate and press Enter Or Draw: Select Draw and press Enter Program Output: H 1 hypothesis test type Value of z-standard normal dist. test statistic P-value of test statistic Calculated value of p-hat statistic Size of random sample Hypothesis Test for Population μ ( σ known ) Select: STAT / TESTS / Z-Test Inpt: Use arrow keys – select Stats μo : the population μ stated in H o σ = the standard deviation of the parent pop. x : the sample mean statistic n : sample size μ ≠ μo < μo > μo (select H 1 test type ) Calculate: Select Calculate and press Enter Or Draw: Select Draw and press Enter Program Output: H 1 hypothesis test type Value of z-standard normal distribution test statistic P-value of test statistic Value of sample mean statistic Size of random sample Hypothesis Test for Population μ ( σ unknown ) Select: STAT / TESTS / T-Test Inpt: Use arrow keys – select Stats μo : the population μ stated in H o x : the sample mean statistic Sx : the sample standard deviation statistic n : sample size μ ≠ μo < μo > μo (select H 1 test type ) Calculate: Select Calculate and press Enter Or Draw: Select Draw and press Enter Program Output: H 1 hypothesis test type Value of t-distribution test statistic P-value of test statistic Value of sample mean statistic Value of the sample standard deviation statistic Size of random sample Test for Difference of Population p's ( p 1 - p 2 ) Select: STAT / TESTS / 2-PropZTest x1: the number of successes in first sample n1: size of first sample x2: the number of successes in the second sample n2: sample size of the second sample p1 ≠ p2 < p2 > p2 (select H 1 test type ) Calculate: Select Calculate and press Enter Or Draw: Select Draw and press Enter Program Output: H 1 hypothesis test type Value of z-standard normal distribution test statistic P-value of test statistic Calculated p-hat of first sample Calculated p-hat of second sample Calculated pooled p-hat statistic of two samples Size of first random sample Size of second random sample 2 Find Equation of Regression Line (y = a + bx) , Sample Correlation Coefficient r and the Coefficient of Determination r2 with the TI 83/84 + graphing calculator. a) Clear lists L 1 and L 2 . b) Enter the x-coordinates in list L 1 . c) Enter the y-coordinates in list L 2 . d) Press STAT / TESTS / LinRegTTest ● Xlist : L 1 ● Ylist : L 2 ● Freq : 1 ● β and ρ : ≠ 0 ● RegEQ : ● Select Calculate and press the ENTER key . Program Output: y = a + bx β ≠ 0 and ρ ≠ 0 β (beta) is a population parameter equal to the true value of the slope of the regression line. ρ (rho) is a population parameter equal to the true value of the correlation coefficient. t = value of test statistic derived from a random sample p = P-value of test statistic degrees of freedom of t-distribution = n - 2 a = y-intercept of the regression line b = slope of the regression line s = standard error where larger values of s indicate increased scattering of points r2 = the coefficient of determination r = the sample correlation coefficient Draw Scatter Plot and Graph Regression Line with TI 83/84+ a) Enter the x-coordinates in list L 1 . b) Enter the y-coordinates in list L 2 . c) Press the MODE button . ● Select NORMAL number display mode ● Select FLOAT and set rounding to 4 decimal places ● Select FUNC graph type ● Select CONNECTED plot type ● Select SEQUENTIAL ● Select REAL number mode ● Select FULL screen mode d) Press the WINDOW button. Set axes scale values (Xmin, Xmax, etc.) to fit scatter plot data. e) Press the STAT PLOT key. ( 2ND and Y= ) ● Set Plot 1 to on and all other plots to off ● Type : Select scatter plot icon (top-row-left) ● Xlist : L 1 ● Ylist : L 2 ● Mark : Select desired style of plot marker . f) Press the Y= button. ● Clear out all equations with the CLEAR key . ● \Y 1 = equation of regression line: bx + a or a + bx g) Press the GRAPH button to view the graph of the regression line. Be sure to record the values of a and b so that you can graph the regression line in the future. Also record the values of r and r2. Linear Regression and Correlation Calculations 5 After the least squares regression line is graphed, points on the regression line can be found as follows: Press the CALC (2ND TRACE) key and select value. Then enter a value for the x variable and press the ENTER key. Continue entering other values of x as desired. Linear Correlation Hypothesis Test H 0 : ρ = 0 - There is no linear correlation. H 1 : ρ ≠ 0 - There is linear correlation. Probability distribution of the test statistic t = r / √( (1 – r2 ) / (n – 2) ) is a t-distribution with n-2 degrees of freedom. Normal Quantile Plot – Check to see if a sample of n data points came from a normal population. Enter the sample data values in list L 1 . Press the STAT PLOT key. ( 2ND and Y= keys) ● Turn Plot1 on and the other plots to off . ● Type : Select the plot icon in row-2-right. ● Data List : L 1 ● Data Axis : X ● Mark : Select the desired data marker style. Press the ZOOM key and 9 to generate a quantile plot of the sample data values. Press the TRACE key to view the x-y coordinates of points on the graph. Warning! Quantile plots of sample data taken from a uniform distribution may appear to be somewhat linear, however, the plot follows a systematic curved pattern about a straight line and therefore it is not considered to be a linear plot. 7 Example 1: The data set below is a random sample of 16 data values taken from an exponential population with μ = 4 and σ = 4. Exponential populations are very skewed to the right and therefore normal quantile plots of samples taken from an exponential population should not be linear. { 0.160, 7.697, 0.552, 2.266, 2.469, 5.254, 8.143, 2.211, 2.346, 3.901, 3.1619 , 3.105, 11.821, 0.737, 1.415, 16.282 } Example 2: The data set below is a random sample of 21 data values taken from a normal population with μ = 100 and σ = 16. Since the sample was taken from a normal population, the normal quantile plot of the data set should be linear or almost linear. See the comment below. { 113.27, 107.32, 67.68, 122.48, 114.57, 87.15, 90.44, 99.49, 120.36, 103.38, 99.26, 110.60, 88.92, 118.99, 92.63, 101.33, 116.60, 110.19, 98.20, 96.79, 124.90 } If the quantile plot is linear or almost linear, the data values most likely came from a normal population. If the quantile plot is not linear, the parent population is probably not normal. z- sc or e of th e pe rc en ti le r an k of x . z- sc or e of th e pe rc en til e ra nk o f x. Comment: The n data values, { x i }, are first sorted in ascending order. Each x i is assigned a percentile rank P i = ( i – 0.5) / n. Each y-coordinate y i = the z-score corresponding to P i = invNorm(P i , 0, 1). You can judge the straightness of a line by eye. Do not pay too much attention to points at the ends of the plot, unless they are quite far from the line. It is common for a few points at either end to stray from the line somewhat. However, a point that is very far from the line when the other points are very close is an outlier, and deserves attention. The Ryan-Joiner test is one of several other tests available to test for normality. 6 Residual Plot – Plot the residuals of a sample of n (x, y) data points associated with the least-squares regression line of the n data points. The residual of a data point (x,y) equals the difference between y and y p = y – y p where y p equals the y-value predicted by the least-squares regression line. A residual equals the unexplained deviation between y and y p . The least-squares regression line minimizes the sum of the squares of the residuals. If the residual plot shows a random pattern about the x-axis, the least-squares regression line is a good model to fit the data. If the residual plot pattern about the x-axis is not random, a nonlinear model such as quadratic or cubic model might be a more appropriate model to fit the data. Select STAT / SetUpEditor and press the ENTER key. Clear L 1 , L 2 and L 3 . Enter the x-coordinates in L 1 . Enter the y-coordinates in L 2 . Now create a special list for the residuals in the list editor. ● Select STAT / EDIT ● Move the edit cursor to the very top of L 3 so that L 3 is highlighted. ● Press the INS key. ( 2nd DEL = the insert key) ● Use the command LIST / NAMES and select RESID from the list and then press the ENTER key to paste in the RESID symbol at the top of the new list. Next calculate the equation of the least-squares regression line and correlation coefficient r . ● Select STAT / TESTS / LinRegTTest ● Xlist: L 1 ● Ylist: L 2 ● Freq: 1 ● β and ρ : ≠ 0 ● RegEQ : ● Select Calculate and press the ENTER key . ● Record the y-intercept a, slope b, r and r2 . ● Select STAT / EDIT to view the residuals which were automatically inserted in list RESID . Now plot the residuals for each x-y data point. ● Press the STAT PLOT key. ( 2ND and Y= keys) ● Turn Plot1 on and the other plots to off . ● Type : Select the scatter plot icon top-row-left. ● Xlist : L 1 ● Ylist : RESID ( Select LIST / NAMES and then select RESID ) ● Mark : Select the desired data marker style. ● Select ZOOM and 9 to generate a residual plot of the sample x-y data pairs. ● Press the TRACE key to view the x-y coordinates of points on the residual plot. Comment: Page 225 of the course text book has two excellent examples that illustrate how to use a residual plot to determine whether or not a linear model is an appropriate model of the data set. This is an important step if you intend to find the equation of the regression line or the correlation coefficient r at a later time.