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SEEING STATISTICS APPLET EXERCISES Applet 1: Influence of a Single Observation on the Median 1.1 When we click on the right-most green dot and drag it so that the speed of the ninth car is 40 mph, the median will be 30.0. 1.2 When we click on the right-most green dot and drag it so that the speed of the ninth car increases from 40 mph to 100 mph, the median will continue to be 30.0. 1.3 The highest possible value for the median is 30.0 mph. The lowest possible median speed is 29.0 mph. 1.4 Even if the ninth car were a jet-powered supersonic vehicle moving at 1000 mph, the median would remain at its maximum possible value: 30.0 mph. On the other hand, the approximate value of the mean would be (1000 + the speeds of the other eight cars)/9, or nearly 140 mph! Of these two sample descriptors, the mean has been more greatly affected by this extreme value in the data. Applet 2: Scatter Diagrams and Correlation 2.1 With the slider in the center position, the best-fit straight line will be horizontal and the coefficient of correlation will be zero. 2.2 When the slider is at the extreme right position, the coefficient of correlation will be +1.0. The best-fit straight line will slope upward and all of the points will fall on the line. 2.3 We position the slider to the right of the center position. Some fine adjustment may be necessary to reach the r = +0.60 value or very close to it. The best-fit line will slope upward and there will be more “scatter” of points above and below the line. 2.4 Clicking the “Switch Sign” button will cause the best-fit line to slope downward, the value of r will now become negative. The amount of scatter of points about the line will be the same as in Applet Exercise 2.3. 2.5 With the slider positioned at the far left, r = -1.0, the best-fit line slopes downward, and all of the points are on the line. As the slider is gradually moved to the far right, the best-fit line “rotates” in a counter-clockwise direction. The value of r approaches 0, becomes positive, and will be + 1.0 when the slider arrives at the far-right position. The amount of scatter increases, then decreases along the way. Applet 3: Sampling 3.1 Results will vary. For example, the sample proportion may have been less than the actual population value 8 times, equal to it 1 time, and greater than it 11 times. 3.2 Again, keep in mind that the results of Applet Exercise 3.1 will vary. Although the average difference would theoretically be expected to be zero, the actual sampling experience may lead to an average difference that is slightly greater than or slightly less than zero. Applet 4: Size and Shape of Normal Distribution 4.1 When we position the top slider at the far left, then gradually move it to the far right, the red distribution moves to the right and its mean gradually increases from approximately -10.0 to approximately +8.0. 4.2 When we use the bottom slider to change the standard deviation so that it is greater than 1, the red curve “flattens out” compared to that of the blue curve. 4.3 Positioning both sliders to the far left, the red curve will move to the extreme left and it will become very narrow. Its approximate mean and standard deviation will be -10.0 and 0.5, respectively, compared to the 0 and 1 values for the standard normal distribution.

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Applet 5: Normal Distribution Areas 5.1 With a mean of 130 hours and a standard deviation of 30 hours, then moving the left boundary to 70 and the right boundary to 190, the probability in the top row will be 0.95. This is the probability that a randomly selected plane will have flown between 70 and 190 hours during the year. In the second row of the display, the values of z corresponding to 70 hours and 190 hours will be z = -2.0 and z = +2.0, respectively. 5.2 With the mean and standard deviation set at 130 and 30, respectively, and the left and right boundaries of the shaded area at 130 and 190, the upper line in the display shows 0.48 as the probability that a randomly selected plane will have flown between 130 and 190 hours during the year. The corresponding values of z are z = 0.0 and z = 2.0. 5.3 With the left and right boundaries at 140 and 170, respectively, the probability that a randomly selected plane will have flown between 140 and 170 hours during the year is shown in the top line of the display as 0.28. The corresponding values of z are z = 0.32 and z = 1.33. 5.4 With the left and right boundaries corresponding to z = 0 and z = +1.0 (or as close to 1.0 as possible), the probability associated with the shaded area is shown as 0.34. Applet 6: Normal Approximation to Binomial Distribution 6.1 With n = 15 and = 0.6, the probability that there are no more than k = 9 females in the sample of 15 walkers is shown in the “Prob” box as 0.5968. The corresponding probability using the normal approximation to the binomial distribution is displayed as 0.6039, a difference of just 0.0071. 6.2 With n = 5 and = 0.6, we find the actual binomial probability that there are no more than k = 3 females in the sample of 5 to be 0.663. The corresponding probability using the normal approximation is 0.676, a difference of 0.013. With the smaller sample size, the normal approximation is a little less close. 6.3 With n = 100 and = 0.6, we find the actual binomial probability that there are no more than k = 60 females in the sample of 100 to be 0.5379. The corresponding probability using the normal approximation is 0.5406, a difference of 0.0027. With the larger sample size, the normal approximation has become a little closer. 6.4

Repeating Applet Exercise 6.1 for k values of 6 through 12, in each case identifying the actual binomial probability that there will be no more than k females in the sample of n = 15:

k 6 7 8 9 10 11 12

binomial probability that x k 0.0950 0.2131 0.3902 0.5968 0.7827 0.9095 0.9729

normal approximation probability that x k 0.0938 0.2146 0.3961 0.6039 0.7854 0.9062 0.9675

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difference, binomial normal approximation 0.0012 -0.0015 -0.0059 -0.0071 -0.0027 0.0033 0.0054

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Applet 7: Distribution of Means: Fair Dice 7.1 Generating 3000 rolls of a single die, using the “Sample Size = 1” applet version. The heights of the six bars will be fairly even, though not perfectly so. With a fair die, such a shape is close to the level distribution we would expect for the distribution of the “means” when each sample is just 1 die. 7.2 Generating 3000 samples, each one representing a set of three dice. The distribution of sample means will now take on more of a symmetric bell shape with a mean that is close to the expected value of 3.5 for samples involving fair dice. 7.3 Generating 3000 samples, each one representing a set of twelve dice. The distribution of sample means will tend to become even more narrow and more symmetrical, again having a mean that is very close to the expected value of 3.5 for samples involving fair dice. 7.4 If there were an additional applet version that allowed each sample to consist of 100 dice, and we generated 2000 of these samples, the distribution of sample means would take on a distribution very close to the normal distribution. According to the central limit theorem, as the sample sizes become larger, the distribution of sample means will approach the normal distribution. Applet 8: Distribution of Means: Loaded Dice 8.1 Generating 3000 rolls of a single loaded die, using the “Sample Size = 1” applet version. The heights of the six bars will be very uneven, as one might expect when the die is weighted to favor one or more of the sides. 8.2 Generating 3000 samples, each one representing a set of three loaded dice. The distribution of sample means will tend to be strongly skewed and will not have a mean that is close to 3.5, the expected mean for samples involving fair dice. 8.3 Generating 3000 samples, each one representing a set of twelve loaded dice. The distribution of sample means will become more narrow, and it will tend to become a little more symmetrical. However, it will continue to have a mean that is not very close to 3.5, the expected mean for samples involving fair dice. 8.4 If there were an additional applet version that allowed each sample to consist of 100 dice, and we generated 2000 of these samples, the distribution of sample means would tend to be relatively normally distributed, even though the underlying distribution of values is decidedly non-normal. Applet 9: Confidence Interval Size 9.1 With the slider positioned so as to specify a 95% confidence interval for , the upper and lower confidence limits are displayed as 1.381 and 1.419, respectively. 9.2 When we move the slider so that the confidence interval is 99%, the confidence interval is now wider. The upper and lower confidence limits are displayed as 1.3751 and 1.4249, respectively. 9.3 When we move the slider so that the confidence interval is 80%, the confidence interval becomes more narrow. 9.4 As we gradually move the slider from the extreme left position to the extreme right position, both the confidence level and the width of the confidence interval increase. Applet 10: Comparing the Normal and Student t Distributions 10.1 With the slider positioned at df = 5, the shape of the t distribution is flatter and wider than that of the standard normal distribution. 10.2 When the slider is moved downward so that df = 2, the t distribution becomes even more flat and wide compared to the shape of the standard normal distribution. 10.3 When the slider is moved upward so that df increases from 2 to 10, the t distribution becomes less flat and less wide, more closely approaching the shape of the standard normal distribution. 10.4 As the slider is moved upward from df = 2 to df = 100, the t distribution becomes less flat and less wide, and it becomes more and more difficult to differentiate between the two distributions. At the df = 100 value, the two curves are practically identical.

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Applet 11: Student t Distribution Areas 11.1 With the slider set so that df = 9 and the left text box containing t = 3.25, the area beneath the curve between t = -3.25 and t = +3.25 is 1.00 - 0.01, or 0.99. 11.2 Gradually moving the slider upward until df = 89, we see that the t value shown in the text box decreases from 3.25 to 2.63. 11.3 We position the slider so that df = 2, then gradually move it upward until df = 100. The t value decreases and the curve becomes more tall and more narrow, approaching the shape of the normal distribution. 11.4 We position the slider so that df = 9, then enter 0.10 into the two-tail probability text box at the right. The t value in the left text box becomes t = 1.83. This corresponds to a right-tail area of 0.05. Referring to the t table that precedes the inside back cover of the book, the corresponding table value of t is t = 1.833. Applet 12: z-Interval and Hypothesis Testing 12.1 When the applet initially loads, the sample mean is displayed as 1.3229 and the 95% confidence interval limits for are displayed as 1.3142 and 1.3316. Based on this confidence interval, it would seem believable that the true population mean might be 1.325 minutes, because this value falls within the limits. 12.2 When we use the slider to increase the sample mean to approximately 1.330 minutes, the 95% confidence interval limits for are displayed as 1.3213 and 1.3386. Based on this confidence interval, it would seem believable that the true population mean might be 1.325 minutes, because this value falls within the limits. 12.3 When we use the slider to decrease the sample mean to approximately 1.310 minutes (e.g., 1.3099 may be the closest you can get), the 95% confidence interval limits are displayed as 1.3013 and 1.3186. Based on this confidence interval, it would not seem believable that the true population mean might be 1.325 minutes, because this value does not fall with the limits. Applet 13: Statistical Power of a Test 13.1 With the left and right sliders set so that = 0.10 and n = 20, we move the bottom slider so that the actual is as close as possible to 10 without being equal to 10. For example, for an actual = 10.01, the power of the test is 0.10. This is the value we would expect, since = 0.10. 13.2 With the left, right, and bottom sliders set so that = 0.05, n = 15, and = the same 10.01 value we selected in Applet Exercise 13.1, we find that moving the left slider upward and downward results in and the power of the test increasing and decreasing together, and that they continue to have the same numerical value. 13.3 With the left and bottom sliders set so that = 0.05 and = 11.2, we move the right slider upward and downward to change the sample size for the test. The power of the test corresponding to each of the selected values of n (2, 10, 20, 40, 60, 80, and 100) are displayed as 0.12, 0.39, 0.67, 0.92, 0.99, 1.0, and 1.0, respectively. 13.4 With the right and bottom sliders set so that n = 20 and = 11.2, we move the left slider upward and downward to change the level for the test. The power of the test corresponding to each of the selected values of (0.01, 0.02, 0.05, 0.10, 0.20, 0.30, 0.40, and 0.50) are displayed as 0.43, 0.53, 0.67, 0.77, 0.87, 0.91, 0.94, and 0.96, respectively.

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Applet 14: Distribution of Difference Between Sample Means 14.1 Setting the top slider so that the difference between the population means is -3.0, the slider at the upper right so that the standard deviation of each population is 2.5, and the slider at the lower right so that n1 = n2 = 20: a. There is quite a lot of overlap between the two population curves at the top of the applet. b. There is practically no overlap at all between the two sampling distribution curves in the center part of the applet. c. Viewing the bottom portion of the applet, and assuming that a sample is going to be taken from each of the two populations, it seems very unlikely that ( x 1 - x 2) will be > 0. Note: In the applet, the red curves represent population 2 (top section) and the sampling distribution of the means from population 2 (center section), respectively. The green curve at the bottom represents the sampling distribution of ( x 1 - x 2). 14.2 Repeating Applet Exercise 14.1, but with the top slider set so the difference between the sample means is +0.5: a. There is so much overlap between the two population curves at the top of the applet that they nearly coincide. b. There is also a considerable amount of overlap between the two sampling distribution curves in the center part of the applet. c. Viewing the bottom portion of the applet, and assuming that a sample is going to be taken from each of the two populations, it quite possible that ( x 1 - x 2) will be > 0. 14.3 Repeating Applet Exercise 14.1, but with the top slider set so the difference between the sample means is 3.0: a. There is quite a lot of overlap between the two population curves at the top of the applet. b. There is practically no overlap at all between the two sampling distribution curves in the center part of the applet. c. Viewing the bottom portion of the applet, and assuming that a sample is going to be taken from each of the two populations, it seems extremely likely ( x 1 - x 2) will be > 0. 14.4 Using the top slider to gradually change the difference between the population means from -0.5 to 4.5: In the set of curves at the top, the underlying population represented by the red curve shifts to the right. In the sampling distributions in the center section, the sampling distribution of means from the red-curve population shifts to the right. In the bottom of the applet, the green curve representing the sampling distribution of ( x 1 - x 2) also shifts to the right. 14.5 Using the upper right slider to gradually increase the population standard deviations from 1.1 to 3.0: This tends to spread out and increase the amount of overlap between the population curves in the top of the applet, to spread out and increase the amount of overlap between the sampling distribution curves in the center of the applet, and to spread out the sampling distribution of ( x 1 - x 2) in the bottom of the applet. 14.6 Using the lower right slider to gradually increase the sample sizes from 2 to 20: There is no change in the underlying populations in the upper portion of the applet. Each of the sampling distribution curves in the center portion of the applet becomes narrower, and the amount of overlap between the two curves tends to decrease. In the bottom portion of the applet, the sampling distribution of ( x 1 - x 2) becomes more narrow.

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Applet 15: F Distribution and ANOVA 15.1 When we use the left slider to increase the number of degrees of freedom for the numerator of the F ratio to df1 = 5, then to df1 = 10, the F distribution curve tends to flatten out slightly and extend further to the right. 15.2 After using the left and right sliders to set the degrees of freedom back to df1 = 2 and df2 = 7, we use the right slider to set the number of degrees of freedom for the denominator to df = 10, then to df = 15. The F distribution curve tends to flatten out and extend further to the right. 15.3 After using the left and right sliders to set the degrees of freedom back to df1 = 2 and df2 = 7, we use the left text box to increase the F value to 9.55. (Be sure to press the Enter or Return key after changing the text box entry.) The probability changes to 0.01. 15.4 After using the left and right sliders to set the degrees of freedom back to df1 = 2 and df2 = 7, we use the left text box to return the F value to 6.54, then the right text box to change the probability to 0.01. The F value is now displayed as 9.57. The sharp-eyed reader will note that the 9.57 F value in this solution differs slightly from the 9.55 F value that was entered in Applet Exercise 15.3. Both correspond to a probability of 0.01 (with the probability rounded to two decimal places). In Applet Exercise 15.3, we entered the exact value F = 9.55 and got a rounded probability of 0.01. In this exercise, we entered the exact probability 0.01 and got a rounded F value of 9.57. Applet 16: Interaction Graph in Two-Way ANOVA 16.1 We center all three sliders so that the value at the far right of each slider scale is a zero. Next, we slide the top slider to the right and set it at +10 to increase the difference between the row means. The two lines representing the row 1 and row 2 effects are parallel, with the “R2” line 10 points above 100 and the “R1” line 10 points below 100. When we slide the top slider further to the right, to +20, the lines remain parallel but the “R2” and “R1” are now at the 120 and 80 levels, respectively. We have made the row 2 effect stronger and the row 1 effect weaker. 16.2 We center all three sliders so that the value at the far right of each slider scale is a zero. Next, we slide the middle slider to the right and set it at +10 to increase the difference between the column means. The line in the graph slopes upward, as we have increased the effect of column 2 and decreased the effect of column 1. When we move this slider further to the right, to +20, the columneffect advantage of column 2 becomes more pronounced, and the line gets even steeper. 16.3 We center all three sliders so that the value at the far right of each slider scale is a zero. Next, we set each of the top two sliders to -20. The lines are parallel and the effect of row 2 is greater than the effect of row 1 (as in Applet Exercise 16.1), and both lines are upward sloping, indicating that the effect of column 2 is greater than the effect of column 1 (as in Applet Exercise 16.2). 16.4 We set each of the top two sliders to their +20 positions, then move the bottom slider across its range of movement, from far left to far right. When the bottom slider is at the far left, the lines are no longer parallel -- there is row-column interaction and the row 2/column 2 combination has the strongest effect (also see the table to the right of the graph). As we move the bottom slider from left to right, the lines rotate in opposite directions. At the far right position of the bottom slider we see that the row 1/column 2 and row 2/column 1 combinations have the strongest effect and the row 1/column 1 combination has the weakest effect. 16.5 We wish to set the sliders so that (a) Row 1 will have a greater effect (higher mean) than Row 2, (b) Column 2 will have a greater effect than Column 1, and (c) the strongest positive interaction effect will be from the combination of Row 1 and Column 2. There are a great many possibilities, but settings of -20, +15, and +30 for the three sliders will result in these conditions being satisfied.

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Applet 17: Chi-Square Distribution 17.1 When we use the slider to increase the number of degrees of freedom, the curve flattens out and extends further to the right. 17.2 Using the “ChiSq” text box, we enter a larger chi-square value. (Be sure to press the Enter or Return key after changing the text box entry.) The associated probability will decrease. 17.3 Using the “Prob” text box, we enter a numerically higher probability. The associated chi-square value will decrease. Applet 18: Regression: Point Estimate for y 18.1 Moving the slider to represent a dexterity test score of 10, we find that the point estimate for productivity is 49.2. This is shown in the graph as well as at the left side of the equation in the upper portion of the applet. 18.2 When we move the slider to increase the dexterity test score to 11, the point estimate for productivity goes up to 52.2 18.3 In Applet Exercise 18.2, we increased the dexterity test score by 1 and the point estimate for productivity went up by 3.0. This is what we would expect, since 3.0 is the slope of the estimation equation. 18.4 If the dexterity test score is increased by 5 points, the point estimate for productivity should increase by the slope multiplied by 5, or by 3(5) = 15 units. We can verify this with the applet: When we use the slider to increase the dexterity test score from 10 to 15, the point estimate for productivity increases from 49.2 to 64.2, an increase of 15 units. Applet 19: Point Insertion Diagram and Correlation 19.1 There are many possibilities. For example, we can just insert three points so that, as we go from left to right, each point is higher than the point inserted before it. In this case, r will be positive. 19.2 Inserting points like those shown in the Applet 19 figure in the text will result in r being between +0.7 and +0.9. We can make r < 0.7 by simply inserting some additional points that are more distant from the best-fit line. 19.3 If we reset to a clear screen and insert two points that form an upward-sloping diagonal line, the value of r will be +1.0. It takes only two points to determine a straight line. 19.4 Resetting to a clear screen, we can simply insert two distant points that form a downward-sloping line, then insert 8 additional points between them that are either on (or nearly on) the line itself. This will result in r being either -1.0 or very close to -1.0. 19.5 Resetting to a clear screen, we can simply insert 10 points that tend to form what is approximately a horizontal line. As a result, r will be either 0 or very close to 0. Applet 20: Regression Error Components 20.1 When we click on the regression line and move it so that it is as close as possible to horizontal, the value of r2 will be 0. In this case, the regression line is explaining 0% of the variation in productivity. 20.2 When we click on the regression line and move it so that the slope of the equation is approximately 1.0, this improves the predictive ability of the equation somewhat and increases the value of r 2 from 0 to approximately 0.5. 20.3 When we click on the regression line and gradually move it so that the slope of the equation increases from approximately 1.0 to approximately 5.0, r 2 will increase at first, and then it will decrease. This is because the best-fit slope is 3.0, and we have moved toward it, matched it, then gone beyond it. 20.4 Clicking on the “Find Best Model” button, we obtain Productivity = 19.2 + 3.0*Dexterity. The slope of the equation is 3.0, indicating that a 1-point higher score on the dexterity test tends to be accompanied by a 3-unit increase in productivity. The r2 = 0.91 that is displayed indicates that 91% of the variation in productivity is explained by the regression equation. 249

Applet 21: Mean Control Chart 21.1 Using the first applet and pressing the “New Sample” button 10 times, we generate 10 samples from the process. Our conclusion will tend to be that the process is in control. 21.2 Continuing with the first applet and pressing the “New Sample” button 90 more times to reach the limit of 100 consecutive samples, we will tend to conclude that the process was in control during the 100 samples. 21.3 Using the second applet and pressing the “New Sample” button 35 times, we generate 35 samples. This is the “ill-behaved” process, and we will very likely find that the process was not in control. The most typical clue will be a sample mean that falls outside the control limits, but there could be other possibilities as well. 21.4 Continuing with the second applet, we press the “New Sample” button 65 more times to reach the limit of 100 consecutive samples. It is very likely that we will find that the process has “settled down” since the samples that were generated in Applet Exercise 21.3. 21.5 Repeating Applet Exercises 21.1 through 21.4, the points in both charts will be different, but the well-behaved process will still tend to be in control and the ill-behaved process will still tend to enter an out-of-control situation somewhere between sample 15 and sample 35.

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