Question

A study was carried out to determine whether the resistance of a control
circuit in a machine is lower when the machine motor is running. To
investigate this question, some of the control circuits were tested as
follows. Their resistance was measured while the machine motor was not
running and then again while the motor was running for a certain period of
time. The values found are listed in ‘Dataset’, with kilo-Ohms
as the unit of measurement. Answer the following questions.
1. Carry out an appropriate statistical test to determine whether the
resistances have decreased.
2. Explain why the test was done with measurements when the motor
was not running first and then measurements with the motor
running, in other words, the order of the measurements.
3. Create a confidence interval for the difference.
4. Setting X as the resistance without and Y as the resistance with the
motor running, set up the least-squares linear regression equation
Y = a + bX
predicting Y from X. Carry out an appropriate test on the
coefficients and on the correlation coefficients.
5. State what values of the coefficients of a linear equation should be
consistent with the model in part (1).

Dataset:

Resistance:
Motor running	Motor not running
12.00	11.93
11.68	11.78
11.44	11.64
11.04	11.53
11.08	11.80
11.46	12.21
11.78	12.43
11.98	12.34
11.42	11.53
10.84	11.19
10.78	11.32
10.30	10.65
10.18	10.63
10.40	10.73

Answer 1

1.

Paired T-Test and CI: Motor running, Motor not running

Paired T for Motor running - Motor not running

                              N     Mean   StDev SE Mean
Motor running         14   11.170   0.608    0.163
Motor not running   14   11.551   0.596    0.159
Difference              14 -0.3807 0.2416   0.0646

95% upper bound for mean difference: -0.2664
T-Test of mean difference = 0 (vs < 0): T-Value = -5.90 P-Value = 0.000

Since p-value<0.05 so there is sufficient evidence to conclude that the resistances have decreased.

2. Since the resistance was measured while the machine motor was not running and then again while the motor was running for a certain period of time and want to determine whether the resistance of a control circuit in a machine is lower when the machine motor is running. For this reason the test was done with measurements when the motor was not running first and then measurements with the motor running i.e. the order of the measurements.

3. 95% CI for mean difference: (-0.5202 kilo-Ohms, -0.2412 kilo-Ohms).

4.

Regression Analysis: Motor running versus Motor not running

The regression equation is
Motor running = 0.32 + 0.939 Motor not running

Predictor            Coef SE Coef     T          P
Constant            0.321    1.339 0.24      0.815
Motor not running 0.9393   0.1158 8.11 0.000

S = 0.248643   R-Sq = 84.6%   R-Sq(adj) = 83.3%

Analysis of Variance

Source               DF      SS      MS      F      P
Regression          1 4.0707 4.0707 65.84 0.000
Residual Error 12 0.7419 0.0618
Total             13 4.8126

Since P-value of ANOVA test and t test of regression coefficient<0.05 so we conclude that X and Y are linearly related.

5.

Predictor              Coef SE Coef     T          P
Constant              0.321    1.339 0.24      0.815
Motor not running 0.9393   0.1158 8.11      0.000