Question

Electrobat, a battery manufacturer, is investigating how storage temperature affects the performance of one of its popular deep-cell battery models used in recreational vehicles. Samples of 30 fully charged batteries were subjected to a light load under each of four different storage temperature levels. The hours until deep discharge (meaning ≤ 20% of charge remaining) were measured. The data is shown in the accompanying table. 0 degrees F 30 degrees F 60 degrees F 90 degrees F (hrs to discharge) (hrs to discharge) (hrs to discharge) (hrs to discharge) 3 6 12 12 5 8 13 15 ⋮ ⋮ ⋮ ⋮ 4 9 9 15 Click here for the Excel Data File a-1. Construct an ANOVA table. (Round intermediate calculations to at least 4 decimal places. Round "SS", "MS", "p-value" to 4 decimal places and "F" to 3 decimal places.) a-2. Specify the competing hypotheses to test whether the temperature levels have different mean discharge times. H0: μ0 deg ≤ μ30 deg ≤ μ60 deg ≤ μ90 deg. HA: Not all population means are equal. H0: μ0 deg = μ30 deg = μ60 deg = μ90 deg. HA: Not all population means are equal. H0: μ0 deg ≥ μ30 deg ≥ μ60 deg ≥ μ90 deg. HA: Not all population means are equal. a-3. At the 5% significance level, what is the conclusion of the test? a-4. What about the 1% significance level? Reject H0 Do not reject H0 b. If significant differences exist, use Tukey’s HSD method at the 5% significance level to determine which temperature levels have different mean discharge times. (You may find it useful to reference the q table). (Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.) b-1. What q-parameter value did you use? 3.68 c. If significant differences exist, repeat analysis b. above using Fisher’s LSD method at the 5% significance level to determine which temperature levels have different mean discharge times. (You may find it useful to reference the t table). Did any difference pairs change? c-1. What t-parameter value did you use? 1

Answer 1

Sol:

Let X1 = 0 degrees ,  X2 = 30 degrees , X3 = 60 degrees , X4 = 90 degrees

Following table shows the calculations:

	X1	X2	X3	X4
	3	6	12	12
	5	8	13	15
	6	8	14	11
	6	7	9	11
	3	6	11	11
	4	6	14	10
	6	9	13	12
	4	8	12	14
	4	9	10	11
	3	8	12	11
	4	9	12	13
	6	6	10	10
	3	9	13	15
	5	7	14	12
	6	7	14	15
	6	9	11	14
	6	9	9	15
	3	9	13	15
	6	8	12	12
	4	9	9	11
	6	7	14	11
	4	7	13	10
	3	6	12	13
	4	7	14	15
	3	9	13	11
	6	6	13	13
	6	9	12	12
	5	8	11	12
	3	7	9	12
	4	9	9	15
Mean	4.5667	7.7333	11.9	12.4667
variance	1.5644	1.3747	2.9897	3.0161

So we have

The grand mean is

So

and

Therefore

----

Since there are 4 different groups so we have k=4. Therefore degree of freedoms are:

-------------

Now

F test statistics is

So p-value of the test is 0.0000. Since P-value is less than 0.05 so we reject the null hypothesis. That is on the basis of sample evidence we can conclude that populations are different.

(b)

Here we have 4 groups and total number of observations are 120. So degree of freedom is

df=120-4= 196

Critical value for , df=196 and k=4 is

So Tukey's HSD will be

Following table shows the Tukey's interval:

groups (i-j)	xbari	xbarj	ni	nj	HSD	xbari-xbarj	Lower limit	Upper limit	Significant(Yes/No)
mu1-mu2	4.5667	7.7333	30	30	1	-3.1666	-4.17	-2.17	Yes
mu1-mu3	4.5667	11.9	30	30	1	-7.3333	-8.33	-6.33	Yes
mu1-mu4	4.5667	12.4667	30	30	1	-7.9	-8.9	-6.9	Yes
mu2-mu3	7.7333	11.9	30	30	1	-4.1667	-5.17	-3.17	Yes
mu2-mu4	7.7333	12.4667	30	30	1	-4.7334	-5.73	-3.73	Yes
mu3-mu4	11.9	12.4667	30	30	1	-0.5667	-1.57	0.43	No