Question

Consider the following situation: Electrical engineers have a device that tests for battery life (in minutes) by placing a battery under a controlled electrical load and measuring how long it lasts. They are interested in comparing the performance of 4 brands of batteries. They replicated the experiment 4 times by randomly assigning a battery brand to be used in the electrical load device each time they measured battery life. In other words, they made 16 ‘runs’ and randomized the order in which the battery brands were used.

The data they obtained was:

BrandA	BrandB	BrandC	BrandD
110	118	108	117
113	116	107	112
108	112	112	115
115	117	108	119

1. State the Null and Alternative Hypothesis in words (both hypotheses) and using statistical notation (null hypothesis only).

2. Compute the means and sample standard errors for the brands. Upload the file. This can be an Excel spreadsheet, a photo/screenshot/scanned image.

3. Compute the sums of squares for Treatment, Error, and Total, and complete the ANOVA table below. For each SS, MS, and F calculation round to nearest whole number. For example, calculate SS > round to nearest whole number. Use that SS to calculate MS > round to nearest whole number. Use that MS to calculate F > round to nearest whole number.

Source	DF	SS	MS	F
Treatment
Error
Total

4. Using the F - table and your ANOVA table results, what is the critical F-value for a test of the hypothesis at the 5% level of significance?

5. Based on your F-test statistic and F-critical value, write a complete conclusion of your hypothesis test.

6. What is the value from the Tukey table you will used to calculate the Tukey comparisons?  

7. Construct mean comparisons using the Tukey method and upload your results. Be sure to provide the Tukey W value (i.e. the value you calculated by which you compared the treatment means) and the letter grouping for each of the treatment means. Based on these Tukey comparisons, what battery brand(s) would you conclude differ in mean lifetime? This upload can be, for example, a photo taken of your work and an upload of that image.

8. Now use Minitab to conduct this analysis. Upload an image of your your Minitab output that shows your ANOVA table and grouping information of Tukey mean comparisons.

Answer 1

SUMMARY
Groups	Count	Sum	Average	Variance	Standard error
BrandA	4	446	111.5	9.6667	3.1091
BrandB	4	463	115.75	6.9167	2.63
BrandC	4	435	108.75	4.9167	2.2174
BrandD	4	463	115.75	8.9167	2.9861

3.

Source of Variation	DF	SS	MS	F
Treatment	3	142	47	6
Error	12	91	8
Total	15	233

4.

Critical value=F_0.05,3,12=3.49

Since F-ratio>3.49 so we reject H₀ at 5% level of significance.

5.

There is sufficient evidence that the mean life for brands are not all equal.

6. Q_0.05,4,12= 4.05

8.

One-way ANOVA: Measurement for battery life versus Brand

Source DF      SS     MS     F      P
Brand    3 141.69 47.23 6.21 0.009
Error   12   91.25   7.60
Total   15 232.94

S = 2.758   R-Sq = 60.83%   R-Sq(adj) = 51.03%

                         Individual 95% CIs For Mean Based on
                         Pooled StDev
Level N    Mean StDev --------+---------+---------+---------+-
A      4 111.50   3.11          (--------*-------)
B      4 115.75   2.63                      (--------*-------)
C      4 108.75   2.22 (--------*-------)
D      4 115.75   2.99                      (--------*-------)
                         --------+---------+---------+---------+-
                             108.5     112.0     115.5     119.0

Pooled StDev = 2.76

Grouping Information Using Tukey Method

Brand N     Mean Grouping
D      4 115.750 A
B      4 115.750 A
A      4 111.500 A B
C      4 108.750    B

Means that do not share a letter are significantly different.

Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons among Levels of Brand

Individual confidence level = 98.83%

Brand = A subtracted from:

Brand   Lower Center   Upper --------+---------+---------+---------+-
B      -1.541   4.250 10.041                  (-------*-------)
C      -8.541 -2.750   3.041        (-------*-------)
D      -1.541   4.250 10.041                  (-------*-------)
                               --------+---------+---------+---------+-
                                    -7.0       0.0       7.0      14.0

Brand = B subtracted from:

Brand    Lower Center   Upper --------+---------+---------+---------+-
C      -12.791 -7.000 -1.209 (-------*-------)
D       -5.791   0.000   5.791            (-------*-------)
                                --------+---------+---------+---------+-
                                     -7.0       0.0       7.0      14.0

Brand = C subtracted from:

Brand Lower Center   Upper --------+---------+---------+---------+-
D      1.209   7.000 12.791                      (-------*-------)
                              --------+---------+---------+---------+-
                                   -7.0       0.0       7.0      14.0