Example

A company would like to compare between three types (brands) of tires, A, B and C based on their ability to maintain the same pressure at leaving them for 5 hrs. For each tire brand, the company also used 15 to test and also used two different sizes; 13 and 15 inches, See the table below. Pressure was initially 32 psi.

Tire A / Tire B
Tire / Size 13 / Size 15 / Size 13 / Size 15
1 / 32 / 30.1 / 25.6 / 28
2 / 31 / 28.5 / 28.5 / 27
3 / 31 / 27.7 / 24.3 / 26.1
4 / 31.7 / 28.5 / 22.8 / 28
5 / 30.9 / 29.3 / 30 / 27.3
6 / 29.8 / 30.3 / 29.9 / 27.9
7 / 30.1 / 29.9 / 24.6 / 28.1
8 / 31.8 / 31.2 / 26.6 / 22.3
9 / 31 / 30.2 / 24.9 / 22.4
10 / 30.8 / 30.4 / 25.7 / 21.9
11 / 31 / 31 / 29.7 / 27.3
12 / 31.7 / 31 / 28.8 / 30.4
13 / 31 / 30.7 / 26.4 / 30.5
14 / 31.3 / 30.5 / 26 / 31
15 / 31 / 30.8 / 25 / 29
  1. Identify the independent and dependent variables and their format?

Independent: Tire type: A or B (nominal)

Size: 13 or 15 (nominal)

Dependent: Pressure (ratio or scale)

  1. Write the null and alternative hypotheses?

Ho1: Tire A average Pressure = Tire B average Pressure

H11: Tire A average Pressure ≠ Tire B average Pressure

Ho2: Size 13 average Pressure = Size 15 average Pressure

H12: Size 13 average Pressure ≠ Size 15 average Pressure

  1. What type of test will be used and why?ANOVA, two independent varaiables.
  2. Conduct the analysis and provide a copy of the summary table?

Tests of Between-Subjects Effects
Dependent Variable: Pressure
Source / Type III Sum of Squares / df / Mean Square / F / Sig.
Corrected Model / 213.286a / 3 / 71.095 / 18.959 / .000
Intercept / 49432.881 / 1 / 49432.881 / 13182.520 / .000
Tire / 202.401 / 1 / 202.401 / 53.975 / .000
Size / .963 / 1 / .963 / .257 / .614
Tire * Size / 9.923 / 1 / 9.923 / 2.646 / .109
Error / 209.993 / 56 / 3.750
Total / 49856.160 / 60
Corrected Total / 423.279 / 59
a. R Squared = .504 (Adjusted R Squared = .477)
  1. Conduct a post-hoc analysis using the Sheffe’s and Tukey’s procedures?

Pairwise Comparisons
Dependent Variable: Pressure
(I) Tire / (J) Tire / Mean Difference (I-J) / Std. Error / Sig.b / 95% Confidence Interval for Differenceb
Lower Bound / Upper Bound
Type A / Type B / 3.673* / .500 / .000 / 2.672 / 4.675
Type B / Type A / -3.673* / .500 / .000 / -4.675 / -2.672
Based on estimated marginal means
*. The mean difference is significant at the .05 level.
b. Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments).
Pairwise Comparisons
Dependent Variable: Pressure
(I) Size / (J) Size / Mean Difference (I-J) / Std. Error / Sig.a / 95% Confidence Interval for Differencea
Lower Bound / Upper Bound
Size 13 / Size 15 / .253 / .500 / .614 / -.748 / 1.255
Size 15 / Size 13 / -.253 / .500 / .614 / -1.255 / .748
Based on estimated marginal means
a. Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments).

Conclusion: There is a significant difference between the two tires average pressure at α = 0.05.

There is no significant difference in the average pressure between the two sizes; 13 and 15.

Tire A is better than tire B in regard to the ability to hold pressure.

The interaction between the tire type and size is not significant.

Recommendations: Use tire A regardless of the size.

Doing at-test to compare between the two sizes:

Group Statistics
Size / N / Mean / Std. Deviation / Std. Error Mean
Pressure / Size 13 / 30 / 28.8300 / 2.80273 / .51171
Size 15 / 30 / 28.5767 / 2.58986 / .47284
Independent Samples Test
Levene's Test for Equality of Variances / t-test for Equality of Means
F / Sig. / t / df / Sig. (2-tailed) / Mean Difference / Std. Error Difference / 95% Confidence Interval of the Difference
Lower / Upper
Pressure / Equal variances assumed / 1.765 / .189 / .364 / 58 / .717 / .25333 / .69672 / -1.14131 / 1.64797
Equal variances not assumed / .364 / 57.642 / .717 / .25333 / .69672 / -1.14149 / 1.64816

Conducting a t-test between the two tire types, A and B

Group Statistics
Tire / N / Mean / Std. Deviation / Std. Error Mean
Pressure / Type A / 30 / 30.5400 / .99779 / .18217
Type B / 30 / 26.8667 / 2.57311 / .46978
Independent Samples Test
Levene's Test for Equality of Variances / t-test for Equality of Means
F / Sig. / t / df / Sig. (2-tailed) / Mean Difference / Std. Error Difference / 95% Confidence Interval of the Difference
Lower / Upper
Pressure / Equal variances assumed / 22.361 / .000 / 7.290 / 58 / .000 / 3.67333 / .50387 / 2.66473 / 4.68194
Equal variances not assumed / 7.290 / 37.529 / .000 / 3.67333 / .50387 / 2.65288 / 4.69378

See this expanded example

Tire A / Tire B / Tire C
Tire / Size 13 / Size 15 / Size 13 / Size 15 / Size 13 / Size 15
1 / 32 / 30.1 / 25.6 / 28 / 32 / 32
2 / 31 / 28.5 / 28.5 / 27 / 32 / 32
3 / 31 / 27.7 / 24.3 / 26.1 / 31.9 / 32
4 / 31.7 / 28.5 / 22.8 / 28 / 31.8 / 32
5 / 30.9 / 29.3 / 30 / 27.3 / 31.7 / 31.9
6 / 29.8 / 30.3 / 29.9 / 27.9 / 31.8 / 31.9
7 / 30.1 / 29.9 / 24.6 / 28.1 / 32 / 31.9
8 / 31.8 / 31.2 / 26.6 / 22.3 / 31.5 / 30.7
9 / 31 / 30.2 / 24.9 / 22.4 / 32 / 32
10 / 30.8 / 30.4 / 25.7 / 21.9 / 31.7 / 32
11 / 31 / 31 / 29.7 / 27.3 / 31.7 / 31.8
12 / 31.7 / 31 / 28.8 / 30.4 / 30.9 / 31.8
13 / 31 / 30.7 / 26.4 / 30.5 / 31.3 / 32
14 / 31.3 / 30.5 / 26 / 31 / 31.4 / 31.9
15 / 31 / 30.8 / 25 / 29 / 31.2 / 31.9
Tests of Between-Subjects Effects
Dependent Variable: Pressure
Source / Type III Sum of Squares / df / Mean Square / F / Sig.
Corrected Model / 400.023a / 5 / 80.005 / 31.541 / .000
Intercept / 79501.000 / 1 / 79501.000 / 31342.697 / .000
Tire / 388.858 / 2 / 194.429 / 76.652 / .000
Size / .245 / 1 / .245 / .097 / .757
Tire * Size / 10.920 / 2 / 5.460 / 2.153 / .123
Error / 213.067 / 84 / 2.537
Total / 80114.090 / 90
Corrected Total / 613.090 / 89
a. R Squared = .652 (Adjusted R Squared = .632)
Pairwise ( post-hoc) Comparisons
Dependent Variable: Pressure
(I) Tire / (J) Tire / Mean Difference (I-J) / Std. Error / Sig.b / 95% Confidence Interval for Differenceb
Lower Bound / Upper Bound
Type A / Type B / 3.673* / .411 / .000 / 2.856 / 4.491
Type C / -1.217* / .411 / .004 / -2.034 / -.399
Type B / Type A / -3.673* / .411 / .000 / -4.491 / -2.856
Type C / -4.890* / .411 / .000 / -5.708 / -4.072
Type C / Type A / 1.217* / .411 / .004 / .399 / 2.034
Type B / 4.890* / .411 / .000 / 4.072 / 5.708
Based on estimated marginal means
*. The mean difference is significant at the .05 level.
b. Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments).
Estimates
Dependent Variable: Pressure
Tire / Mean / Std. Error / 95% Confidence Interval
Lower Bound / Upper Bound
Type A / 30.540 / .291 / 29.962 / 31.118
Type B / 26.867 / .291 / 26.288 / 27.445
Type C / 31.757 / .291 / 31.178 / 32.335