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BENG 2143 – Assignment
ANOVA & Regression
Question 2 (a) - 10 marks:
It is suspected that the experimental temperature in which batteries are activated affects their life. Fifty homogenous
batteries were tested, ten at each of five temperatures, and the data (activated life in seconds) are in the Excel file in the
sheet entitle ‘Question 2 (a)’. Based on the data, answer the following questions.
a) Test at the 0.05 level of significance to indicate whether the mean of the batteries life are differ significantly for the
five tested temperatures.
b) Construct a 95 percent confidence interval for the mean of batteries life at 75℃.
Solution:
a) The ANOVA table:
!
"
$
%
&
"
: At least two of the means are not equal.
From the ANOVA table, test statistic, 𝐹 = 257. 2678
Critical point, 𝐹
!.!&,%,)&
The test statistics, 𝐹 = 257. 2678 falls in the rejection region. The null hypothesis is rejected.
Conclusion: The mean of the batteries life for the five tested temperatures are significantly different.
( 6 marks)
b) 95 percent confidence interval for the mean of batteries life at 75℃;
The point estimate of the mean at 75℃ is 𝑦 1
*&℃
Thus, 95% confidence interval for 𝜇
*&℃
!.!#&,&
( ")
)
$.#$
#!
ANOVA
Source of Variation SS df MS F P-value
Between Groups 3323.9 4 830.975 257.2678 4.75756E-
Within Groups 306.85 95 3.
Total 3630.75 99
Question 2 (b) - 10 marks:
Referring to the Excel file in the sheet entitle ‘Question 2 (b)’. The data are values of pressure (psi) in a torsion spring for
five settings of the angle between the legs of the spring is a free position. The setting angles are; 67°, 71°, 75°, 79° and
83° and 65 values of pressure are recorded. Based on the given data,
a) Find the mean for pressure of the torsion spring for each angel. What can you conclude about the values gain?
b) Test at the 0.05 level of significance that the average of pressure in a torsion spring is the same for all five setting of
the angles.
Solution:
a) By using Microsoft Excel, the mean for each angle is as in the table below.
Angle 67 ° 71 ° 75 ° 79 ° 83 °
Mean pressure (psi) 83.50 85.3333 88.08 89.6364 91.
Based on the values of the mean calculated, it can be said that the pressure of the torsion spring is getting higher parallel
with the increase of the angle between the legs of the spring.
( 4 marks)
b)The ANOVA table:
!
.*°
*"°
*&°
*)°
0$°
"
: At least two of the means are not equal.
From the ANOVA table, test statistic, 𝐹 = 97. 7883
Critical point, 𝑝 − 𝑣𝑎𝑙𝑢𝑒 = 1. 4 × 10
1 #&
As the 𝑝 − 𝑣𝑎𝑙𝑢𝑒 = 1. 4 × 10
1 #&
< 0. 05 , thus reject 𝐻
!
Conclusion: The average of pressure in a torsion spring for five setting of the angles are not all the same.
ANOVA
Source of Variation SS df MS F P-value
Between Groups 367.7446 4 91.9361 97.7883 1.4007E-
Within Groups 56.4093 60 0.
Total 424.1538 64
Observation
Fluidizing gas flow
rate (lb/hr) - x Heat transfer
𝑥
𝑦
- 1 41.85 69.69 1751.4225 4856.6961 2916. efficiency - y - 2 75.77 113.46 5741.0929 12873.1716 8596. - 3 99.63 113.54 9926.1369 12891.3316 11311. - 4 101.63 118.75 10328.6569 14101.5625 12068. - 5 100.76 119.72 10152.5776 14332.8784 12062. - 6 107.7 168.38 11599.29 28351.8244 18134. - 7 105.34 169.85 11096.5156 28849.0225 17891. - 8 116.86 170.89 13656.2596 29203.3921 19970. - 9 120.65 171.31 14556.4225 29347.1161 20668.
- 10 124.34 171.43 15460.4356 29388.2449 21315.
- 11 127.98 171.59 16378.8804 29443.1281 21960.
- 12 135.38 171.63 18327.7444 29456.8569 23235.
- 13 146.78 171.93 21544.3684 29559.9249 25235.
- 14 150.22 173.92 22566.0484 30248.1664 26126.
- 15 154.19 221.44 23774.5561 49035.6736 34143.
- 16 157.65 300.32 24853.5225 90192.1024 47345.
- 17 160.86 295.23 25875.9396 87160.7529 47490.
- 18 162.93 290.12 26546.1849 84169.6144 47269.
- 19 164.28 299.16 26987.9184 89496.7056 49146.
- 20 165.21 301.45 27294.3441 90872.1025 49802.
- 21 166.23 300.34 27632.4129 90204.1156 49925.
- 22 172.45 303.04 29739.0025 91833.2416 52259.
- 23 175.91 302.45 30944.3281 91476.0025 53203.
- 24 178.13 301.97 31730.2969 91185.8809 53789.
- 25 180.05 300.32 32418.0025 90192.1024 54072.
- 26 185.23 307.55 34310.1529 94587.0025 56967.
- 27 193.77 308.12 37546.8129 94937.9344 59704.
- 28 200.54 310.88 40216.2916 96646.3744 62343.
- 29 213.81 355.69 45714.7161 126515.3761 76050.
- 30 215.34 370.12 46371.3156 136988.8144 79701.
- 31 217.41 378.02 47267.1081 142899.1204 82185.
- 32 220.63 375.19 48677.5969 140767.5361 82778.
- 33 222.25 376.32 49395.0625 141616.7424 83637.
- 34 227.89 380.48 51933.8521 144765.0304 86707.
- 35 234.54 420.65 55009.0116 176946.4225 98659.
- 36 245.55 502.34 60294.8025 252345.4756 123349.
- 37 256.76 498.13 65925.6976 248133.4969 127899.
- 38 260.33 500.11 67771.7089 250110.0121 130193.
- 39 266.98 551.54 71278.3204 304196.3716 147250.
- 40 270.02 562.12 72910.8004 315978.8944 151783.
- 41 273.89 569.95 75015.7321 324843.0025 156103.
- 42 276.19 573.22 76280.9161 328581.1684 158317.
- 43 280.32 580.33 78579.3024 336782.9089 162678.
- 44 281.65 583.78 79326.7225 340799.0884 164421.
- 45 284.33 600.91 80843.5489 361092.8281 170856.
- 46 287.96 650.88 82920.9616 423644.7744 187427.
- 47 290.44 673.45 84355.3936 453534.9025 195596.
- 48 295.65 682.44 87408.9225 465724.3536 201763.
- 49 301.22 690.45 90733.4884 476721.2025 207977.
- 50 305.02 705.77 93037.2004 498111.2929 215273.
- Sum 9700.50 17800.37 2114007.7988 7945991.7373 4049572.
- Mean 194.0100 356.
Question 3 (b):
An engineer at a semiconductor company wants to model the relationship between the device gain or hFE
and two
parameters; emitter-RS (𝑥
"
) and base-RS (𝑥
). The data consists of nineteen readings for hFE and both parameters are
in the Excel file in the sheet entitle ‘Question 3 (b)’.
a) Fit a multiple linear regression to the data.
b) Predict hFE when 𝑥
"
= 14 and 𝑥
c) Test the null-hypothesis that neither emitter-RS nor base-RS are linearly related to the hFE. Use 𝛼 = 0. 05.
Solution:
a) Calculation using Excel:
Calculate using formula,
22
3 !
3 !
3 "
3 "
3 !
2
3 "
2
3 !
3 "
"
The fitted model: 𝑦 = 88. 1898 − 53. 2960 𝑥
"
(5 marks)
b) When 𝑥
"
= 14 and 𝑥
(1 marks)
c) The ANOVA table;
Solution:
!
"
"
4
≠ 0 , for at least one value of j.
Emitter-RS
x
cBase-RS
x
hFE-1M-5V
y
14.62 226.0 128.40 213.7444 51076.000 16486.560 1877.208 29018.400 3304.
15.63 220.0 52.62 244.2969 48400.000 2768.864 822.451 11576.400 3438.
14.62 217.4 113.90 213.7444 47262.760 12973.210 1665.218 24761.860 3178.
15.00 220.0 98.01 225.0000 48400.000 9605.960 1470.150 21562.200 3300.
14.50 226.5 139.90 210.2500 51302.250 19572.010 2028.550 31687.350 3284.
15.25 224.1 102.60 232.5625 50220.810 10526.760 1564.650 22992.660 3417.
16.12 220.5 48.14 259.8544 48620.250 2317.460 776.017 10614.870 3554.
15.13 223.5 109.60 228.9169 49952.250 12012.160 1658.248 24495.600 3381.
15.50 217.6 82.68 240.2500 47349.760 6835.982 1281.540 17991.168 3372.
15.13 228.5 112.60 228.9169 52212.250 12678.760 1703.638 25729.100 3457.
15.50 230.2 97.52 240.2500 52992.040 9510.150 1511.560 22449.104 3568.
16.12 226.5 59.06 259.8544 51302.250 3488.084 952.047 13377.090 3651.
15.13 226.6 111.80 228.9169 51347.560 12499.240 1691.534 25333.880 3428.
15.63 225.6 89.09 244.2969 50895.360 7937.028 1392.477 20098.704 3526.
15.38 234.0 171.90 236.5444 54756.000 29549.610 2643.822 40224.600 3598.
15.50 230.0 66.80 240.2500 52900.000 4462.240 1035.400 15364.000 3565.
14.25 224.30 157.10 203.0625 50310.490 24680.410 2238.675 35237.530 3196.
14.50 240.50 208.40 210.2500 57840.250 43430.560 3021.800 50120.200 3487.
14.62 223.70 133.40 213.7444 50041.690 17795.560 1950.308 29841.580 3270.
SUM 288.13 4285.50 2083.52 4374.71 967181.97 259130.61 31285.29 472476.30 64980.
MEAN 15.16 225.55 109.
