A manufacturer has tested 11 lithium-ion batteries, and finds that their lifetime is 400 charge cycles with a standard deviation of 10. An engineer is testing a cheaper design and wants to know if it has changed battery life. The average lifetime for 10 of the new batteries is 390 cycles, and the standard deviation of that sample is 8. Define change in lifetime as New minus Old. What is the best estimate of the change in lifetime

To estimate the change in battery lifetime for the new design compared to the old one, we should perform a hypothesis test. However, since we are asked for the best estimate of the change in lifetime rather than conducting a formal test, we can calculate the difference between the means of the two groups—the new design and the old one.

The best estimate of the change in lifetime (Δ) would be:

Δ = (average lifetime of new batteries) - (average lifetime of old batteries)

Plugging in the values given:

Δ = 390 cycles (new design) - 400 cycles (old design)

Δ = -10 cycles

So, the best estimate of the change in lifetime would be a decrease of 10 cycles.

Extra: When assessing the difference between two means (in this case, the lifetimes of two different designs of batteries), we typically look at the mean of each group. Here, the old battery has a mean lifetime of 400 cycles and the new battery has a mean lifetime of 390 cycles. The mean lifetime of the new batteries is less than that of the old ones, suggesting that the new design might have a shorter lifespan.

It's also important to consider the variability of the measurements, usually expressed as the standard deviation. In the case of the old batteries, the standard deviation is 10, and for the new batteries, it is 8. This tells us that the lifetimes of the old batteries are more spread out from their mean than the new batteries. The standard deviation helps us understand the consistency of the battery life in each group.

However, to make a definitive statement about the change in lifetime and whether it is statistically significant, one would usually conduct a t-test or similar statistical test that takes into account both the means and the standard deviations, as well as the sample sizes of each group. This test would offer insights into whether the observed difference in means is likely due to natural variation or if it is statistically significant, meaning it's probably due to the change in the design of the batteries. Additionally, a 95% confidence interval could be calculated to determine the range within which the true mean change in battery lifetime is likely to fall with 95% confidence.