Question

4. A gear manufacturer tests the gear pitch uniformity on 4 gears of a larger production run The gears have 7 teeth each (one measurement per tooth). The results are below. Tooth Pitch, Gear 1 (mm) Pitch, Gear 2 (mm) Pitch, Gear 3 (mm) 24.9 24.2 24.8 25.7 25.2 25.2 24.6 24.9 25,5 25.2 25.0 24.2 25.1 25.1 25.0 25.2 25.1 25.3 25.5 24.7 25.1 Pitch, Gear 4(mm) 24.9 24.4 25.1 24.4 24.3 25.2 25.3 a) Determine the overall sample mean of the test Solution 4a b) Determine variance of the test. Solution 4b c) Determine the 95 % confidence interval for the standard deviation of the gear pitch. Solution 4c

Answer 1

Tooth	Pitch, Gear 1 (mm)	Pitch, Gear 2 (mm)	Pitch, Gear 3 (mm)	Pitch, Gear 4 (mm)	Average
1	24.9	24.9	25.0	24.9	24.925
2	24.2	25.5	25.2	24.4	24.825
3	24.8	25.2	25.1	25.1	25.05
4	25.7	25.0	25.3	24.4	25.1
5	25.2	24.2	25.5	24.3	24.8
6	25.2	25.1	24.7	25.2	25.05
7	24.6	25.1	25.1	25.3	25.025
				Overall Average	24.97

1 Tooth Pitch, Gear 1 (mm Pitch, Gear 2 (mm Pitch, Gear 3 (mm Pitch, Gear 4 (mmAverage 24.9 24.2 24.8 25.7 25.2 25.2 24.6 24.9 25.5 25.2 25.0 24.2 25.1 25.1 25.0 25.2 25.1 25.3 25.5 24.7 25.1 24.9 24.4 25.1 24.4 24.3 25.2 25.3 Oea Average 24.925 24.825 25.05 25.1 24.8 25.05 25.025 24.97 4 6 10 Variance 0.1556 12 13

EXCEL FORMULA for Variance =VAR(B2:E8)

a) Overall sample mean of the test = Sum of all 28 observations / 28 = 699.1/28 = 24.97

b) Variance of the test = 0.1556

c) For 95% confidence interval, we first need to find the critical values.

Degrees of freedom = n-1 = 28-1 = 27,

$\alpha/2$ = (1-0.95)/2 = 0.025

1-$\alpha/2$ = 1-0.025 = 0.075

$\chi^2_{1-\alpha/2}} = \chi^2_{0.975,27}$ = 14.573   (refer this values from TABLE IV below)

$\chi^2_{\alpha/2}} = \chi^2_{0.025,27}$ = 43.194 (refer this values from TABLE IV below)

Confidence interval is:

$\inline \frac{(n-1)s^2}{\chi^2_{\alpha/2}} < \sigma^2 < \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}$, where, variance of the test, s² = 0.1556

$\inline \frac{(28-1)0.1556^2}{43.194} < \sigma^2 < \frac{(28-1)0.1556^2}{14.573}$

0.0973 < $\inline \sigma^2$ < 0.2883

0.3119 < $\inline \sigma$ < 0.5369

TABLE IV Chl-Square (2 Distribution Area to the Right of Critical Value Degrees of Freedom 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 6.635 9210 9.348 11.345 948811.143 13.277 9.23611.071 12.833 15.086 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 2558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 3.053 3.816 4575 5.578 17.275 19.675 21.920 24.725 3571 440 5.226 6.304 18.54921.026 23.337 26.217 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 5.229 6.262 .2 8.547 22.307 24.996 27488 30.578 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 8.260 9.591 10.851 12.443 28412 31.410 34.170 37.566 8897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 9542 10.982 12.338 14.042 30813 33.924 36.781 40.289 10.196 1.689 13.091 14.848 32.007 35.172 38.076 41.638 10856 12.40 13.848 15.659 33.196 36.415 39.364 42.980 11524 3.120 14.611 16.473 34.382 37.652 40.646 44.314 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 12.879 14.573 16.151 18.114 36.741 40.113 43.194 46.963 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 14.257 16.047 17.708 19.768 39.087 42.557 45.722 49.588 14.954 16.791 18.493 20.599 40.256 43.773 46.979 50.892 0.016 0.103 0.211 0.115 0.216 0.352 0.584 1.064 0554 083 1145 1610 0.001 0.004 2.706 4.605 6.251 7.779 3.841 5.991 7.815 5.024 7.378 0.020 00 0.297 0484 0.711 10 12 13 14 15 16 18 20 21 23 24 25 27 28 30