TABLE 4. POPULATION OF 00 VALUES WITH µ = 6. AND = 7.5 8. 6.4 0. 9.9 9.8 6.6 6. 5.7 5. 6.3 6.7 30.6.6.3 30.0 6.5 8. 5.6 0.3 35.5.9 30.7 3.. 9. 6. 6.8 5.3 4.3 4.4 9.0 5.0 9.9 5. 0.8 9.0.9 5.4 7.3 3.4 38..6 8.0 4.0 9.4 7.0 3.0 7.3 5.3 6.5 3.5 8.0.4 3.4. 7.7 7. 7.0 5. 4.0 4.5 3.8 8. 6.8 7.7 39.8 9.8 9.3 8.5 4.7.0 8.4 6.4 4. 9.9.8 36.0.3 8.8.8 8.5 30.9 9. 8. 30.3 6.5 6.9 6.6 8. 4. 5.5 30. 8.9 8.9 7.6 9.6 7.9 4.9.3 6.7 PLATE 4-
CLASS EXPERIMENT: Randomly select data sets of 0 values and compute mean and variance for the sample. Set : 4. 4.4 8.5 5.3 3. 9.6 3.9.3 4.0 6.5 Set : 33.9.3.3 5. 8.9 9.6 8.5 36.0 7. 30.6 Set 3: 8.0. 8.9 33. 30. 6.5 5. 9.0.8 6.3 Set 4: 3. 30.0 4. 8.9 7..4.3.3 6.4 4.5 Set 5: 5.4 5..3 3..6.3 5.7.4 3. 5.3 Set 6: 3. 8.9 7.0 0.8 0.3 8.4 3.5 6.8 33. 7.3 Set 7:.0 5.3 6.5 3. 5.4 8.5.7 4. 5.5 7.3 Set 8: 30.3 0.3 0.9.8 9. 3. 5.3 30.9 9.4 8.0 Set 9:.3 5.6 5.8 4.7 8.9 30..3 5. 7.9 5.7 Set 0:.3 3.0.3 3. 30.0 4.0 6.8 9.0 30.6 6.8 Solution: Set : Set : Set 3: Set 4: Set 5: Set 6: Set 7: Set 8: Set 9: Set 0: y = 5.89 y = 6.4 y = 6.03 y = 3.84 y = 4.45 y = 6.64 y = 5.96 y = 4.0 y = 5.66 y = 6.49 S = 8.4 S = 36.9 S = 9.39 S =.05 S = 0.35 S = 7.7 S = 8.65 S = 9.5 S = 8.37 S = 5.9 PLATE 4-
CLASS EXPERIMENT: Randomly select data sets of 0 values and compute mean and variance for the sample. Set : 3.9 3.0 35.5 5.6 7.3 5. 8.9.9 0. 5.7 9.8 4.0 9. 8.8 7. 5.5 5.3.4 6..7 Set : 8.4 5. 4. 8. 3.4.0.5 6.8 6.4 5.5 35.5 7.3 0. 9.3 8.9 9.8 7. 0.3 3.5 8. Set 3: 3.3.3 8.0.8 5.3 4. 30.9 7.0.4 0.9 3.9 5.4 6..3 4..9 9.0 6.8 3.8 3.5 Set 4: 6.8.4.0 30.3 5.6.9 5..8 4.4 7. 33. 30.0.6 4. 4.0 9.8 0.3 6. 9. 6.4 Set 5: 3.3 9.9.5 9.3 5..4..9 5.4 30. 8.5 3. 38. 7.6 4.9.3 5.3 0.8 6.8 36.0 Set 6: 3. 5.4. 8.9 8.4 4.4.8.3 6.3 6.8 6. 5. 3.8 7. 8.5 30. 4.0.6 4. 6.8 Set 7:. 5.3 8.4 4.0 7.0 6.4 8.9 7.3 5. 8.8 0.3.9 9..6 9.9 0.9 7..5 8.5 9. Set 8: 4.5 30.9. 33.9 6.7 5. 8.5.8 4.0 8. 6.5 9.6 30.0 30.3 6. 9.3.3 3.8.9 5.7 Set 9: 0.8 9.3 5.5 4.5 6.7 9..4 7. 6. 6. 6.4 9.4 8.5 3. 6.8 3.0.5 9.8 8. 5.3 Set 0: 8.5 30.9.4 36.0 4.0 5.4 6. 30.0 3..9 6.8 7. 5.5.6 8.9 30.7 7.6 30. 0.3 5. Solution: Set : y = 5.30 S = 4.37 Set 6: y = 3.86 S =.90 Set : y = 5.48 S = 0.55 Set 7: y = 3.78 S = 5.6 Set 3: y = 5.50 S =.7 Set 8: y = 5.57 S = 8.56 Set 4: y = 5.60 S = 37.3 Set 9: y = 3.95 S = 9.40 Set 5: y = 4.53 S = 69.54 Set 0: y = 6.67 S = 8.0 PLATE 4-3
CHANGES IN ESTIMATES FOR POPULATION MEAN AND VARIANCE WITH INCREASING SAMPLE SIZE # S y 0 6.9 8. 0 5.9.9 30 5.9 0.0 40 6.5 8.6 50 6.6 0.0 60 6.4 7.6 70 6.3 7. 80 6.3 8.4 90 6.3 7.8 PLATE 4-4
CHI SQUARE DISTRIBUTION: v S,v. Distribution of sample variances. Used to build confidence intervals for population variance. 3. Starts at 0 and goes to 4. Distribution is NOT symmetric 5. Based on degrees of freedom, v, from sample., v 6. Table gives critical value which delineates area in upper tail. PLATE 4-5
STUDENT or t DISTRIBUTION: t y µ / n t. Distribution of sample means.. Used to build confidence intervals for population mean when (n < 30). 3. Starts at and goes to +. 4. Like normal distribution, t distribution is symmetric. 5. Based on degrees of freedom, v, from sample. 6. Table gives critical t value which delineates area in upper tail. PLATE 4-6
F Distribution: F S S. Distribution of ratio of two sample variances.. Used to build confidence interval for ratio of two population variances. 3. Starts at 0 and goes to +. F(, v, v ) 4. Based number of degrees of freedom from sample. 5. Distribution is not symmetric, however: F (, v, v ) F (, v, v ) 6. Table gives critical F value which delineates area in upper tail. PLATE 4-7
A CONFIDENCE INTERVAL FOR A POPULATION MEAN USING A SAMPLE MEAN & STANDARD DEVIATION Lower Tail / - Confidence Region -t +t Upper Tail / µ y ± t ( /, v ) S n OR y t ( /, v) S n µ y t ( /, v) S n PLATE 4-8
EXAMPLE A sample of 0 circle readings has a mean of 34.5", and a standard deviation of ±.", what is the: a) 95% confidence interval for the pop. mean? b) 99% confidence interval for the pop. mean? c) would a measurement of 35.7 be acceptable for this set of data? Part a) Step : = 0.05 ( - 0.95) so / = 0.05 v = 0 - = 9 Look up critical value of t =.093 (0.05, 9) Step : 33.5 34.5.093. 0 µ 34.5.093. 0 35.5 PLATE 4-9
Part b: 99% CONFIDENCE INTERVAL Step : = 0.0 ( - 0.99) so / = 0.005 v = 0 - = 9 Look up critical value of t (0.005, 9) =.86 Step : 33. 34.5.86. 0 µ 35.8.86. 0 35.8 Note that the 99% confidence interval is larger than the 95%. This interval indicates that 99% of the time the population mean is between 33. and 35.8. PLATE 4-0
TABLE D-3 Critical Values for t Distribution v 0.400 0.350 0.300 0.50 0.00 0.50 0.00 0.050 0.05 0.00 0.005 0.00 0.0005 0.35 0.50 0.77.000.376.963 3.078 6.34.705 3.86 63.639 38.0 636.8 0.89 0.445 0.67 0.86.06.386.886.90 4.303 6.964 9.95.40 3.579 3 0.77 0.44 0.584 0.765 0.978.50.638.353 3.83 4.54 5.84 0.6.954 4 0.7 0.44 0.569 0.74 0.94.90.533.3.776 3.748 4.604 6.897 8.60 5 0.67 0.408 0.559 0.77 0.90.56.476.05.57 3.365 4.03 5.895 6.880 6 0.65 0.404 0.553 0.78 0.906.34.440.943.447 3.43 3.708 5.08 5.96 7 0.63 0.40 0.549 0.7 0.896.9.45.895.365.998 3.500 4.785 5.408 8 0.6 0.399 0.546 0.706 0.889.08.397.860.306.896 3.356 4.50 5.04 9 0.6 0.398 0.543 0.703 0.883.00.383.833.6.8 3.50 4.304 4.78 0 0.60 0.397 0.54 0.700 0.879.093.37.8.8.764 3.69 4.49 4.605 0.60 0.396 0.540 0.697 0.876.088.363.796.0.78 3.06 4.09 4.45 0.59 0.395 0.539 0.695 0.873.083.356.78.79.68 3.055 3.933 4.39 3 0.59 0.394 0.538 0.694 0.870.079.350.77.60.650 3.0 3.854 4.30 4 0.58 0.393 0.537 0.69 0.868.076.345.76.45.64.977 3.789 4.48 5 0.58 0.393 0.536 0.69 0.866.074.34.753.3.60.947 3.734 4.079 6 0.58 0.39 0.535 0.690 0.865.07.337.746.0.583.9 3.688 4.0 7 0.57 0.39 0.534 0.689 0.863.069.333.740.0.567.898 3.647 3.970 8 0.57 0.39 0.534 0.688 0.86.067.330.734.0.55.878 3.6 3.96 9 0.57 0.39 0.533 0.688 0.86.066.38.79.093.539.86 3.580 3.887 0 0.57 0.39 0.533 0.687 0.860.064.35.75.086.58.845 3.553 3.853 0.57 0.39 0.53 0.686 0.859.063.33.7.080.58.83 3.58 3.8 0.56 0.390 0.53 0.686 0.858.06.3.77.074.508.89 3.506 3.795 3 0.56 0.390 0.53 0.685 0.858.060.39.74.069.500.807 3.486 3.770 4 0.56 0.390 0.53 0.685 0.857.059.38.7.064.49.797 3.467 3.748 5 0.56 0.390 0.53 0.684 0.856.058.36.708.060.485.787 3.45 3.77 6 0.56 0.390 0.53 0.684 0.856.058.35.706.056.479.779 3.435 3.708 7 0.56 0.389 0.53 0.684 0.855.057.34.703.05.473.77 3.4 3.69 8 0.56 0.389 0.530 0.683 0.855.056.33.70.048.467.763 3.409 3.675 9 0.56 0.389 0.530 0.683 0.854.055.3.699.045.46.756 3.397 3.66 30 0.56 0.389 0.530 0.683 0.854.055.30.697.04.457.750 3.385 3.647 35 0.55 0.388 0.59 0.68 0.85.05.306.690.030.438.74 3.340 3.59 40 0.55 0.388 0.59 0.68 0.85.050.303.684.0.43.704 3.307 3.55 60 0.54 0.387 0.57 0.679 0.848.045.96.67.000.390.660 3.3 3.46 0 0.54 0.386 0.56 0.677 0.845.04.89.658.980.358.67 3.60 3.374 0.53 0.385 0.55 0.675 0.84.037.8.645.960.36.576 3.9 3.300 PLATE 4-
Constructing Confidence Intervals for First Sample Set : y = 5.89 S = 8.4 Set : y = 6.4 S = 36.9 Set 3: y = 6.03 S = 9.39 Set 4: y = 3.84 S =.05 Set 5: y = 4.45 S = 0.35 Set 6: y = 6.64 S = 7.7 Set 7: y = 5.96 S = 8.65 Set 8: y = 4.0 S = 9.5 Set 9: y = 5.66 S = 8.37 Set 0: y = 6.49 S = 5.9 Construct a 90% confidence interval for µ. Does the µ of 6. lie in the interval? SET : 3.40 < µ < 8.38 SET 6: 3.6 < µ < 9.67 SET :.75 < µ < 9.73 SET 7: 4.6 < µ < 7.66 SET 3: 3.48 < µ < 8.58 SET 8:.45 < µ < 6.57 SET 4:. < µ < 6.56 SET 9: 4.74 < µ < 6.57 SET 5:.59 < µ < 6.3 SET 0: 4. < µ < 8.76 PLATE 4-
SELECTING A SAMPLE SIZE Specifications on a project call for 95% of all angles to be measured to within ±.." How many turnings should be measured if the standard error in a single angle is ±4"? Note: must use standard normal distribution. n t / I n.960 4.0. 3.9 4 PLATE 4-3
CAN INTERVAL BE DECREASED BEYOND REASON? Assume estimated error in angle is ±8", plot n versus size of interval. Interval 400 00 000 800 600 400 00 0 Interval vs Repetitions 3 5 7 9 3 5 Repetitions Note that gains in precision stabilize at 8 repetitions. (4DR) PLATE 4-4
A CONFIDENCE INTERVAL FOR A POPULATION VARIANCE USING A SAMPLE VARIANCE / - /,v Confidence Region /,v / v S /,v < < v S /,v NOTE: Since distribution is not symmetric, critical values must be determined for both lower and upper tails. PLATE 4-5
EXAMPLE A sample of 0 circle readings has a standard deviation of ±.", what is the: a) 95% confidence interval for the pop. variance? b) 99% confidence interval for the pop. variance? Part a) Step : = 0.05 ( - 0.95) so / = 0.05, v = 0 - = 9 Look up critical value of and (0.05, 9) (0.975, 9) Step :.55 9 (.) 3.85 < < 9 (.) 8.9 9.40 Part b: 99% CONFIDENCE INTERVAL Step : = 0.0 ( - 0.99) so / = 0.005, v = 0 - = 9 Look up critical value of (0.005, 9) and (0.995, 9) Step :.7 9 (.) 38.58 < < 9 (.) 6.84.5 Note that the 99% confidence interval is larger than the 95%. This interval indicates that 99% of the time the population variance is between.7 and 8.9. PLATE 4-6
Table D- Critical Values for Distribution v 0.999 0.995 0.990 0.975 0.950 0.900 0.500 0.00 0.050 0.05 0.00 0.005 0.00 0.00000 0.000039 0.00057 0.00098 0.004 0.06 0.455.705 3.84 5.03 6.634 7.877 0.8 0.00 0.0 0.0 0.05 0.0 0..39 4.6 5.99 7.38 9. 0.60 3.8 3 0.0 0.07 0. 0. 0.35 0.58.37 6.5 7.8 9.35.34.84 6.6 4 0.09 0. 0.30 0.48 0.7.06 3.36 7.78 9.49.4 3.8 4.86 8.47 5 0. 0.4 0.55 0.83.5.6 4.35 9.4.07.83 5.09 6.75 0.5 6 0.38 0.68 0.87.4.64.0 5.35 0.64.59 4.45 6.8 8.55.46 7 0.60 0.99.4.69.7.83 6.35.0 4.07 6.0 8.48 0.8 4.3 8 0.86.34.65.8.73 3.49 7.34 3.36 5.5 7.53 0.09.96 6. 9.5.74.09.70 3.33 4.7 8.34 4.68 6.9 9.0.67 3.59 7.88 0.48.6.56 3.5 3.94 4.87 9.34 5.99 8.3 0.48 3. 5.9 9.59.83.60 3.05 3.8 4.58 5.58 0.34 7.8 9.68.9 4.7 6.76 3.6. 3.07 3.57 4.40 5.3 6.30.34 8.55.03 3.34 6. 8.30 3.9 3.6 3.57 4. 5.0 5.89 7.04.34 9.8.36 4.74 7.69 9.8 34.53 4 3.04 4.08 4.66 5.63 6.57 7.79 3.34.06 3.68 6. 9.4 3.3 36. 5 3.48 4.60 5.3 6.6 7.6 8.55 4.34.3 5.00 7.49 30.58 3.80 37.70 6 3.94 5.4 5.8 6.9 7.96 9.3 5.34 3.54 6.30 8.85 3.00 34.7 39.5 7 4.4 5.70 6.4 7.56 8.67 0.09 6.34 4.77 7.59 30.9 33.4 35.7 40.79 8 4.9 6.7 7.0 8.3 9.39 0.86 7.34 5.99 8.87 3.53 34.8 37.6 4.3 9 5.4 6.84 7.63 8.9 0..65 8.34 7.0 30.4 3.85 36.9 38.58 43.8 0 5.9 7.43 8.6 9.59 0.85.44 9.34 8.4 3.4 34.7 37.57 40.00 45.3 6.45 8.03 8.90 0.8.59 3.4 0.34 9.6 3.67 35.48 38.93 4.40 46.80 6.98 8.64 9.54 0.98.34 4.04.34 30.8 33.9 36.78 40.9 4.80 48.7 3 7.53 9.6 0.0.69 3.09 4.85.34 3.0 35.7 38.08 4.64 44.8 49.73 4 8.09 9.89 0.86.40 3.85 5.66 3.34 33.0 36.4 39.36 4.98 45.56 5.8 5 8.65 0.5.5 3. 4.6 6.47 4.34 34.38 37.65 40.65 44.3 46.93 5.6 6 9..6.0 3.84 5.38 7.9 5.34 35.56 38.89 4.9 45.64 48.9 54.05 7 9.80.8.88 4.57 6.5 8. 6.34 36.74 40. 43.9 46.96 49.64 55.48 8 0.39.46 3.56 5.3 6.93 8.94 7.34 37.9 4.34 44.46 48.8 50.99 56.89 9 0.99 3. 4.6 6.05 7.7 9.77 8.34 39.09 4.56 45.7 49.59 5.34 58.30 30.59 3.79 4.95 6.79 8.49 0.60 9.34 40.6 43.77 46.98 50.89 53.67 59.70 35 4.69 7.9 8.5 0.57.47 4.80 34.34 46.06 49.80 53.0 57.34 60.7 66.6 40 7.9 0.7.6 4.43 6.5 9.05 39.34 5.8 55.76 59.34 63.69 66.77 73.40 50 4.67 7.99 9.7 3.36 34.76 37.69 49.33 63.7 67.50 7.4 76.5 79.49 86.66 60 3.74 35.53 37.48 40.48 43.9 46.46 59.33 74.40 79.08 83.30 88.38 9.95 99.6 0 77.76 83.85 86.9 9.57 95.70 00.6 9.33 40.3 46.57 5. 58.95 63.65 73.6 PLATE 4-7
CONSTRUCT CONFIDENCE INTERVALS FOR POPULATION VARIANCE Recalling first data set: Set : Set : Set 3: Set 4: Set 5: Set 6: Set 7: Set 8: Set 9: Set 0: y = 5.89 y = 6.4 y = 6.03 y = 3.84 y = 4.45 y = 6.64 y = 5.96 y = 4.0 y = 5.66 y = 6.49 S = 8.4 S = 36.9 S = 9.39 S =.05 S = 0.35 S = 7.7 S = 8.65 S = 9.5 S = 8.37 S = 5.9 Construct a 90% confidence interval for. Does the 7.5 lie in the interval? of SET : 9.80 < < 49.78 SET 6: 4.5 < < 73.70 SET : 9.30 < < 98.08 SET 7: 4.60 < < 3.38 SET 3: 0.3 < < 5.4 SET 8: 0.38 < < 5.76 SET 4:.73 < < 59.59 SET 9: 4.45 < <.6 SET 5: 5.5 < < 7.97 SET 0: 8.3 < < 4.3 PLATE 4-8
Table D- Critical Values for Distribution v 0.999 0.995 0.990 0.975 0.950 0.900 0.500 0.00 0.050 0.05 0.00 0.005 0.00 0.00000 0.000039 0.00057 0.00098 0.004 0.06 0.455.705 3.84 5.03 6.634 7.877 0.8 0.00 0.0 0.0 0.05 0.0 0..39 4.6 5.99 7.38 9. 0.60 3.8 3 0.0 0.07 0. 0. 0.35 0.58.37 6.5 7.8 9.35.34.84 6.6 4 0.09 0. 0.30 0.48 0.7.06 3.36 7.78 9.49.4 3.8 4.86 8.47 5 0. 0.4 0.55 0.83.5.6 4.35 9.4.07.83 5.09 6.75 0.5 6 0.38 0.68 0.87.4.64.0 5.35 0.64.59 4.45 6.8 8.55.46 7 0.60 0.99.4.69.7.83 6.35.0 4.07 6.0 8.48 0.8 4.3 8 0.86.34.65.8.73 3.49 7.34 3.36 5.5 7.53 0.09.96 6. 9.5.74.09.70 3.33 4.7 8.34 4.68 6.9 9.0.67 3.59 7.88 0.48.6.56 3.5 3.94 4.87 9.34 5.99 8.3 0.48 3. 5.9 9.59.83.60 3.05 3.8 4.58 5.58 0.34 7.8 9.68.9 4.7 6.76 3.6. 3.07 3.57 4.40 5.3 6.30.34 8.55.03 3.34 6. 8.30 3.9 3.6 3.57 4. 5.0 5.89 7.04.34 9.8.36 4.74 7.69 9.8 34.53 4 3.04 4.08 4.66 5.63 6.57 7.79 3.34.06 3.68 6. 9.4 3.3 36. 5 3.48 4.60 5.3 6.6 7.6 8.55 4.34.3 5.00 7.49 30.58 3.80 37.70 6 3.94 5.4 5.8 6.9 7.96 9.3 5.34 3.54 6.30 8.85 3.00 34.7 39.5 7 4.4 5.70 6.4 7.56 8.67 0.09 6.34 4.77 7.59 30.9 33.4 35.7 40.79 8 4.9 6.7 7.0 8.3 9.39 0.86 7.34 5.99 8.87 3.53 34.8 37.6 4.3 9 5.4 6.84 7.63 8.9 0..65 8.34 7.0 30.4 3.85 36.9 38.58 43.8 0 5.9 7.43 8.6 9.59 0.85.44 9.34 8.4 3.4 34.7 37.57 40.00 45.3 6.45 8.03 8.90 0.8.59 3.4 0.34 9.6 3.67 35.48 38.93 4.40 46.80 6.98 8.64 9.54 0.98.34 4.04.34 30.8 33.9 36.78 40.9 4.80 48.7 3 7.53 9.6 0.0.69 3.09 4.85.34 3.0 35.7 38.08 4.64 44.8 49.73 4 8.09 9.89 0.86.40 3.85 5.66 3.34 33.0 36.4 39.36 4.98 45.56 5.8 5 8.65 0.5.5 3. 4.6 6.47 4.34 34.38 37.65 40.65 44.3 46.93 5.6 6 9..6.0 3.84 5.38 7.9 5.34 35.56 38.89 4.9 45.64 48.9 54.05 7 9.80.8.88 4.57 6.5 8. 6.34 36.74 40. 43.9 46.96 49.64 55.48 8 0.39.46 3.56 5.3 6.93 8.94 7.34 37.9 4.34 44.46 48.8 50.99 56.89 9 0.99 3. 4.6 6.05 7.7 9.77 8.34 39.09 4.56 45.7 49.59 5.34 58.30 30.59 3.79 4.95 6.79 8.49 0.60 9.34 40.6 43.77 46.98 50.89 53.67 59.70 35 4.69 7.9 8.5 0.57.47 4.80 34.34 46.06 49.80 53.0 57.34 60.7 66.6 40 7.9 0.7.6 4.43 6.5 9.05 39.34 5.8 55.76 59.34 63.69 66.77 73.40 50 4.67 7.99 9.7 3.36 34.76 37.69 49.33 63.7 67.50 7.4 76.5 79.49 86.66 60 3.74 35.53 37.48 40.48 43.9 46.46 59.33 74.40 79.08 83.30 88.38 9.95 99.6 0 77.76 83.85 86.9 9.57 95.70 00.6 9.33 40.3 46.57 5. 58.95 63.65 73.6 PLATE 4-9
TABLE D-3 Critical Values for t Distribution v 0.400 0.350 0.300 0.50 0.00 0.50 0.00 0.050 0.05 0.00 0.005 0.00 0.0005 0.35 0.50 0.77.000.376.963 3.078 6.34.705 3.86 63.639 38.0 636.8 0.89 0.445 0.67 0.86.06.386.886.90 4.303 6.964 9.95.40 3.579 3 0.77 0.44 0.584 0.765 0.978.50.638.353 3.83 4.54 5.84 0.6.954 4 0.7 0.44 0.569 0.74 0.94.90.533.3.776 3.748 4.604 6.897 8.60 5 0.67 0.408 0.559 0.77 0.90.56.476.05.57 3.365 4.03 5.895 6.880 6 0.65 0.404 0.553 0.78 0.906.34.440.943.447 3.43 3.708 5.08 5.96 7 0.63 0.40 0.549 0.7 0.896.9.45.895.365.998 3.500 4.785 5.408 8 0.6 0.399 0.546 0.706 0.889.08.397.860.306.896 3.356 4.50 5.04 9 0.6 0.398 0.543 0.703 0.883.00.383.833.6.8 3.50 4.304 4.78 0 0.60 0.397 0.54 0.700 0.879.093.37.8.8.764 3.69 4.49 4.605 0.60 0.396 0.540 0.697 0.876.088.363.796.0.78 3.06 4.09 4.45 0.59 0.395 0.539 0.695 0.873.083.356.78.79.68 3.055 3.933 4.39 3 0.59 0.394 0.538 0.694 0.870.079.350.77.60.650 3.0 3.854 4.30 4 0.58 0.393 0.537 0.69 0.868.076.345.76.45.64.977 3.789 4.48 5 0.58 0.393 0.536 0.69 0.866.074.34.753.3.60.947 3.734 4.079 6 0.58 0.39 0.535 0.690 0.865.07.337.746.0.583.9 3.688 4.0 7 0.57 0.39 0.534 0.689 0.863.069.333.740.0.567.898 3.647 3.970 8 0.57 0.39 0.534 0.688 0.86.067.330.734.0.55.878 3.6 3.96 9 0.57 0.39 0.533 0.688 0.86.066.38.79.093.539.86 3.580 3.887 0 0.57 0.39 0.533 0.687 0.860.064.35.75.086.58.845 3.553 3.853 0.57 0.39 0.53 0.686 0.859.063.33.7.080.58.83 3.58 3.8 0.56 0.390 0.53 0.686 0.858.06.3.77.074.508.89 3.506 3.795 3 0.56 0.390 0.53 0.685 0.858.060.39.74.069.500.807 3.486 3.770 4 0.56 0.390 0.53 0.685 0.857.059.38.7.064.49.797 3.467 3.748 5 0.56 0.390 0.53 0.684 0.856.058.36.708.060.485.787 3.45 3.77 6 0.56 0.390 0.53 0.684 0.856.058.35.706.056.479.779 3.435 3.708 7 0.56 0.389 0.53 0.684 0.855.057.34.703.05.473.77 3.4 3.69 8 0.56 0.389 0.530 0.683 0.855.056.33.70.048.467.763 3.409 3.675 9 0.56 0.389 0.530 0.683 0.854.055.3.699.045.46.756 3.397 3.66 30 0.56 0.389 0.530 0.683 0.854.055.30.697.04.457.750 3.385 3.647 35 0.55 0.388 0.59 0.68 0.85.05.306.690.030.438.74 3.340 3.59 40 0.55 0.388 0.59 0.68 0.85.050.303.684.0.43.704 3.307 3.55 60 0.54 0.387 0.57 0.679 0.848.045.96.67.000.390.660 3.3 3.46 0 0.54 0.386 0.56 0.677 0.845.04.89.658.980.358.67 3.60 3.374 0.53 0.385 0.55 0.675 0.84.037.8.645.960.36.576 3.9 3.300 PLATE 4-0
CONFIDENCE INTERVAL FOR RATIO OF TWO POPULATION VARIANCES - confidence region F l F u / P F l < S S < F u F ( /, v, v ) < S S < F ( /, v, v ) F ( /, v, v ) < S S < F ( /, v, v ) S S F ( /, v, v ) < < S S F ( /, v, v ) S S F ( /, v, v ) < < F ( /, v, v ) S S PLATE 4-
CONFIDENCE INTERVAL FOR RATIO OF TWO POPULATION VARIANCES Determine the confidence interval for the ratio of two population variances based on the two sample set variances. S S F ( /,v,v ) < < S S F ( /,v,v ) EXAMPLE: On Day, 0 EDM distance measurements result in a variance of 5 mm. On Day, additional measurements of the same distance results in a variance of 6 mm. What is the 95% confidence interval for the ratio of the population variances? Since in similar measurement conditions, the expected ratio of the variances is, i.e.,. From the constructed interval is this true? PLATE 4-
Solution: ) / = 0.05, ) Find critical values for F (0.05,9,0) =.84 & F (0.05,0,9) = 3.67 3) Construct interval: 0.30 5 6.84 < < 5 3.67 3.3 6 4) Since is contained in interval, there is no reason to believe that at a 95% level of confidence. PLATE 4-3
Table D-4 (continued) Critical Values for the F Distribution, = 0.05 v v 3 4 5 6 7 8 9 0 5 0 4 30 40 60 0 647.8 799.5 864 899.6 9.8 937. 948. 956.7 963.3 968.6 976.7 984.9 993. 997. 00 006 00 04 38.5 39.00 39.7 39.5 39.30 39.33 39.36 39.37 39.39 39.40 39.4 39.43 39.45 39.46 39.46 39.47 39.48 39.48 3 7.44 6.04 5.44 5.0 4.88 4.73 4.6 4.54 4.47 4.4 4.34 4.5 4.7 4. 4.08 4.04 3.99 3.95 4. 0.65 9.98 9.60 9.36 9.0 9.07 8.98 8.90 8.84 8.75 8.66 8.56 8.5 8.46 8.4 8.36 8.3 5 0.0 8.43 7.76 7.39 7.5 6.98 6.85 6.76 6.68 6.6 6.5 6.43 6.33 6.8 6.3 6.8 6. 6.07 6 8.8 7.6 6.60 6.3 5.99 5.8 5.70 5.60 5.5 5.46 5.37 5.7 5.7 5. 5.07 5.0 4.96 4.90 7 8.07 6.54 5.89 5.5 5.9 5. 4.99 4.90 4.8 4.76 4.67 4.57 4.47 4.4 4.36 4.3 4.5 4.0 8 7.57 6.06 5.4 5.05 4.8 4.65 4.53 4.43 4.36 4.30 4.0 4.0 4.00 3.95 3.89 3.84 3.78 3.73 9 7. 5.7 5.08 4.7 4.48 4.3 4.0 4.0 4.03 3.96 3.87 3.77 3.67 3.6 3.56 3.5 3.45 3.39 0 6.94 5.46 4.83 4.47 4.4 4.07 3.95 3.85 3.78 3.7 3.6 3.5 3.4 3.37 3.3 3.6 3.0 3.4 6.7 5.6 4.63 4.8 4.04 3.88 3.76 3.66 3.59 3.53 3.43 3.33 3.3 3.7 3. 3.06 3.00.94 6.55 5.0 4.47 4. 3.89 3.73 3.6 3.5 3.44 3.37 3.8 3.8 3.07 3.0.96.9.85.79 3 6.4 4.97 4.35 4.00 3.77 3.60 3.48 3.39 3.3 3.5 3.5 3.05.95.89.84.78.7.66 4 6.30 4.86 4.4 3.89 3.66 3.50 3.38 3.9 3. 3.5 3.05.95.84.79.73.67.6.55 5 6.0 4.76 4.5 3.80 3.58 3.4 3.9 3.0 3. 3.06.96.86.76.70.64.59.5.46 6 6. 4.69 4.08 3.73 3.50 3.34 3. 3. 3.05.99.89.79.68.63.57.5.45.38 7 6.04 4.6 4.0 3.66 3.44 3.8 3.6 3.06.98.9.8.7.6.56.50.44.38.3 8 5.98 4.56 3.95 3.6 3.38 3. 3.0 3.0.93.87.77.67.56.50.44.38.3.6 9 5.9 4.5 3.90 3.56 3.33 3.7 3.05.96.88.8.7.6.5.45.39.33.7.0 0 5.87 4.46 3.86 3.5 3.9 3.3 3.0.9.84.77.68.57.46.4.35.9..6 5.83 4.4 3.8 3.48 3.5 3.09.97.87.80.73.64.53.4.37.3.5.8. 5.79 4.38 3.78 3.44 3. 3.05.93.84.76.70.60.50.39.33.7..4.08 3 5.75 4.35 3.75 3.4 3.8 3.0.90.8.73.67.57.47.36.30.4.8..04 4 5.7 4.3 3.7 3.38 3.5.99.87.78.70.64.54.44.33.7..5.08.0 5 5.69 4.9 3.69 3.35 3.3.97.85.75.68.6.5.4.30.4.8..05.98 6 5.66 4.7 3.67 3.33 3.0.94.8.73.65.59.49.39.8..6.09.03.95 7 5.63 4.4 3.65 3.3 3.08.9.80.7.63.57.47.36.5.9.3.07.00.93 8 5.6 4. 3.63 3.9 3.06.90.78.69.6.55.45.34.3.7..05.98.9 9 5.59 4.0 3.6 3.7 3.04.88.76.67.59.53.43.3..5.09.03.96.89 30 5.57 4.8 3.59 3.5 3.03.87.75.65.57.5.4.3.0.4.07.0.94.87 50 5.34 3.97 3.39 3.05.83.67.55.46.38.3...99.93.87.80.7.64 60 5.9 3.93 3.34 3.0.79.63.5.4.33.7.7.06.94.88.8.74.67.58 80 5. 3.86 3.8.95.73.57.45.35.8...00.88.8.75.68.60.5 0 5.5 3.80 3.3.89.67.5.39.30..6.05.94.8.76.69.6.53.43 PLATE 4-4
Table D-4 (continued) Critical Values for the F Distribution, = 0.05 v v 3 4 5 6 7 8 9 0 5 0 4 30 40 60 0 647.8 799.5 864 899.6 9.8 937. 948. 956.7 963.3 968.6 976.7 984.9 993. 997. 00 006 00 04 38.5 39.00 39.7 39.5 39.30 39.33 39.36 39.37 39.39 39.40 39.4 39.43 39.45 39.46 39.46 39.47 39.48 39.48 3 7.44 6.04 5.44 5.0 4.88 4.73 4.6 4.54 4.47 4.4 4.34 4.5 4.7 4. 4.08 4.04 3.99 3.95 4. 0.65 9.98 9.60 9.36 9.0 9.07 8.98 8.90 8.84 8.75 8.66 8.56 8.5 8.46 8.4 8.36 8.3 5 0.0 8.43 7.76 7.39 7.5 6.98 6.85 6.76 6.68 6.6 6.5 6.43 6.33 6.8 6.3 6.8 6. 6.07 6 8.8 7.6 6.60 6.3 5.99 5.8 5.70 5.60 5.5 5.46 5.37 5.7 5.7 5. 5.07 5.0 4.96 4.90 7 8.07 6.54 5.89 5.5 5.9 5. 4.99 4.90 4.8 4.76 4.67 4.57 4.47 4.4 4.36 4.3 4.5 4.0 8 7.57 6.06 5.4 5.05 4.8 4.65 4.53 4.43 4.36 4.30 4.0 4.0 4.00 3.95 3.89 3.84 3.78 3.73 9 7. 5.7 5.08 4.7 4.48 4.3 4.0 4.0 4.03 3.96 3.87 3.77 3.67 3.6 3.56 3.5 3.45 3.39 0 6.94 5.46 4.83 4.47 4.4 4.07 3.95 3.85 3.78 3.7 3.6 3.5 3.4 3.37 3.3 3.6 3.0 3.4 6.7 5.6 4.63 4.8 4.04 3.88 3.76 3.66 3.59 3.53 3.43 3.33 3.3 3.7 3. 3.06 3.00.94 6.55 5.0 4.47 4. 3.89 3.73 3.6 3.5 3.44 3.37 3.8 3.8 3.07 3.0.96.9.85.79 3 6.4 4.97 4.35 4.00 3.77 3.60 3.48 3.39 3.3 3.5 3.5 3.05.95.89.84.78.7.66 4 6.30 4.86 4.4 3.89 3.66 3.50 3.38 3.9 3. 3.5 3.05.95.84.79.73.67.6.55 5 6.0 4.76 4.5 3.80 3.58 3.4 3.9 3.0 3. 3.06.96.86.76.70.64.59.5.46 6 6. 4.69 4.08 3.73 3.50 3.34 3. 3. 3.05.99.89.79.68.63.57.5.45.38 7 6.04 4.6 4.0 3.66 3.44 3.8 3.6 3.06.98.9.8.7.6.56.50.44.38.3 8 5.98 4.56 3.95 3.6 3.38 3. 3.0 3.0.93.87.77.67.56.50.44.38.3.6 9 5.9 4.5 3.90 3.56 3.33 3.7 3.05.96.88.8.7.6.5.45.39.33.7.0 0 5.87 4.46 3.86 3.5 3.9 3.3 3.0.9.84.77.68.57.46.4.35.9..6 5.83 4.4 3.8 3.48 3.5 3.09.97.87.80.73.64.53.4.37.3.5.8. 5.79 4.38 3.78 3.44 3. 3.05.93.84.76.70.60.50.39.33.7..4.08 3 5.75 4.35 3.75 3.4 3.8 3.0.90.8.73.67.57.47.36.30.4.8..04 4 5.7 4.3 3.7 3.38 3.5.99.87.78.70.64.54.44.33.7..5.08.0 5 5.69 4.9 3.69 3.35 3.3.97.85.75.68.6.5.4.30.4.8..05.98 6 5.66 4.7 3.67 3.33 3.0.94.8.73.65.59.49.39.8..6.09.03.95 7 5.63 4.4 3.65 3.3 3.08.9.80.7.63.57.47.36.5.9.3.07.00.93 8 5.6 4. 3.63 3.9 3.06.90.78.69.6.55.45.34.3.7..05.98.9 9 5.59 4.0 3.6 3.7 3.04.88.76.67.59.53.43.3..5.09.03.96.89 30 5.57 4.8 3.59 3.5 3.03.87.75.65.57.5.4.3.0.4.07.0.94.87 50 5.34 3.97 3.39 3.05.83.67.55.46.38.3...99.93.87.80.7.64 60 5.9 3.93 3.34 3.0.79.63.5.4.33.7.7.06.94.88.8.74.67.58 80 5. 3.86 3.8.95.73.57.45.35.8...00.88.8.75.68.60.5 0 5.5 3.80 3.3.89.67.5.39.30..6.05.94.8.76.69.6.53.43 PLATE 4-5
EXAMPLE FOR CLASS PRACTICE Recalling first data set: Set : S = 8.4 Set 6: S 6 = 7.7 Set : S = 36.9 Set 7: S 7 = 8.65 Set 3: S 3 = 9.39 Set 8: S 8 = 9.5 Set 4: S 4 =.05 Set 9: S 9 = 8.37 Set 5: S 5 = 0.35 Set 0: S 0 = 5.9 Construct 95% confidence intervals for the following ratios: ) ) 3) 4) 3 4 5 6 5) 6) 7) 8) 7 8 9 0 9) 0) ) ) 9 5 9 PLATE 4-6
Solutions to plate 4-3.. (0.3,.05). (0.4, 3.83) 3. (0.,3.37) 4. (0.44, 7.7) 5. (0.7,.7) 6. (0.53, 8.58) 7. (0.3, 3.80) 8. (0.55, 8.87) 9. (0.30, 4.85) 0. (0.49, 7.94).(.07, 7.5)*. (0.30, 4.98) Note that set does not contain. Thus there is reason to believe that samples and 9 are not from the same population at a 95% level of confidence. This assumptions is obviously wrong, and thus the test has resulted in an incorrect results, which can be expected 5% of the time. PLATE 4-7