A laptop computer manufacturer found that by improving their battery design, they could increase the time that the laptop computer could be used on a single charge. To verify this improvement, they compared a sample of 10 laptops installed with the improved batteries to a control group of 10 laptops installed with current batteries. After subjecting each laptop to a predetermined set of operations until it ceased to operate, they found that the average lifespan of the improved battery was 4.31 hours, and that the average lifespan of the current battery was 3.68 hours. Correspondingly, the sample standard deviation for the improved batteries was 0.17 hours for the improved group and the sample standard deviation for the current group was 0.22 hours.
Assuming that the two groups have unequal variances, calculate the t-statistic that tests the null hypothesis that the mean lifespan of the two types are equal against the alternative hypothesis that the mean lifespan for the improved batteries is greater than that for the current batteries.
a)1.62
b)7.166
c)-3.682
d)6.907
e)4.923
Welch's t-test for unequal variances:
t = (mean1 - mean2)/√(s^2/n1 + s^2/n2)
t = (4.31 - 3.68)/√(0.17^2/10 + 0.22^2/10)
t = 0.63/0.08792 = 7.166 (rounded)
To calculate the t-statistic, we can use the formula:
t = (x1 - x2) / √[(s1^2 / n1) + (s2^2 / n2)]
Where:
- x1 and x2 are the sample means of the two groups
- s1 and s2 are the sample standard deviations of the two groups
- n1 and n2 are the sample sizes of the two groups
In this case:
- x1 = 4.31 (average lifespan of improved batteries)
- x2 = 3.68 (average lifespan of current batteries)
- s1 = 0.17 (sample standard deviation of improved batteries)
- s2 = 0.22 (sample standard deviation of current batteries)
- n1 = 10 (sample size of improved batteries)
- n2 = 10 (sample size of current batteries)
Substituting these values into the formula, we get:
t = (4.31 - 3.68) / √[(0.17^2 / 10) + (0.22^2 / 10)]
Calculating this expression, we get:
t = 0.63 / √[0.000289 + 0.000484]
Simplifying further, we get:
t = 0.63 / √0.000773
t ≈ 7.166 (rounded to three decimal places)
Therefore, the correct answer is b) 7.166.